Optimal Transport: A Semi-Discrete Approach 9783110635485, 9783110633122

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Gershon Wolansky Optimal Transport

De Gruyter Series in Nonlinear Analysis and Applications

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Editor-in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Manuel del Pino, Santiago de Chile, Chile Avner Friedman, Columbus, Ohio, USA Mikio Kato, Tokyo, Japan Wojciech Kryszewski, Torun, Poland Umberto Mosco, Worcester, Massachusetts, USA Simeon Reich, Haifa, Israel Vicenţiu D. Rădulescu, Krakow, Poland

Volume 37

Gershon Wolansky

Optimal Transport |

A Semi-Discrete Approach

Mathematics Subject Classification 2010 Primary: 52A99, 49J45, 28B20; Secondary: 91-01, 49M29 Author Prof. Dr. Gershon Wolansky The Technion Israel Institute of Technology Department of Mathematics Amado Mathematics Building Technion City 32000 Haifa Israel [email protected]

ISBN 978-3-11-063312-2 e-ISBN (PDF) 978-3-11-063548-5 e-ISBN (EPUB) 978-3-11-063317-7 ISSN 0941-813X Library of Congress Control Number: 2020947504 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2021 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface The theory of optimal transport was born towards the end of the eighteenth century, its founding father being Gaspard Monge [35]. Optimal transport theory has connections with partial differential equations (PDEs), kinetic theory, fluid dynamics, geometric inequalities, probability, and many other mathematical fields as well as with computer science and economics. As such, it has attracted many leading mathematicians in the last decades. There are several very good textbooks and monographs on the subject. For the novice we recommend, as an appetizer, the first book of C. Villani [48], titled “Topics in optimal Transport.” This book describes, in a vivid way, most of what was known on this subject on its publication date (2003). For a dynamical approach we recommend the book of Ambrosio, Gigli, and Savare [2], dealing with paths of probability measures and the vector-field generating them. This fits well with the thesis of Alessio Figalli on optimal transport and action minimizing measures [16]. The main treat is, undoubtedly, the second monster book [49] of Villani, published in 2008. This book emphasizes the geometric point of view and contains a lot more. For the reader interested in application to economics we recommend the recent book [18] of A. Galichon, while for those interested in connections with probability theory and random processes we recommend the book of Rachev and Rüschendorf [38]. As a dessert we recommend the recent book of F. Santambrogio [41], which provides an overview of the main landmarks from the point of view of applied mathematics, and includes also a description of several up-to-date numerical methods. In between these courses the reader may browse through a countless number of review papers and monographs, written by leading experts in this fast growing field. In the current book I suggest an off-road path to the subject. I tried to avoid prior knowledge of analysis, PDE theory, and functional analysis as much as possible. Thus I concentrate on discrete and semi-discrete cases, and always assume compactness for the underlying spaces. However, some fundamental knowledge of measure theory and convexity is unavoidable. In order to make it as self-contained as possible I included an appendix with some basic definitions and results. I believe that any graduate student in mathematics, as well as advanced undergraduate students, can read and understand this book. Some chapters can also be of interest for experts. It is important to emphasize that this book cannot replace any of the books mentioned above. For example, the very relevant subject of elliptic and parabolic PDEs (the Monge–Ampere and the Fokker–Planck equations, among others) is missing, along with regularity issues and many other subjects. It provides, however, an alternative way to the understanding of some of the basic ideas behind optimal transport and its applications and, in addition, presents some extensions which cannot be found elsewhere. In particular, the subject of vector transport, playing a major role in Part II

https://doi.org/10.1515/9783110635485-201

VI | Preface of this book is, to the best of my knowledge, new. The same can be said about some applications discussed in Chapter 8 and Part III. Starting with the most fundamental, fully discrete problem I attempted to place optimal transport as a particular case of the celebrated stable marriage problem. From there we proceed to the partition problem, which can be formulated as a transport from a continuous space to a discrete one. Applications to information theory and game theory (cooperative and non-cooperative) are introduced as well. Finally, the general case of transport between two compact measure spaces is introduced as a coupling between two semi-discrete transports.

Contents Preface | V How to read this book? | XI Notations | XIII 1 1.1 1.2 1.3 1.3.1 1.4 1.5

Introduction | 1 The fully discrete case | 1 Many to few: partitions | 4 Optimal transport in a nutshell | 6 Unbalanced transport | 9 Vector-valued transport and multipartitions | 10 Cooperative and non-cooperative partitions | 13

Part I: Stable marriage and optimal partitions 2 2.1 2.1.1 2.2 2.3 2.4 2.5 2.6

The stable marriage problem | 17 Marriage without sharing | 17 Gale–Shapley algorithm | 18 Where money comes in... | 19 Marriage under sharing | 20 General case | 21 Stability by fake promises | 23 The discrete Monge problem | 27

3 3.1 3.1.1 3.2

Many to few: stable partitions | 31 A non-transferable partition problem | 31 The Gale–Shapley algorithm for partitions | 33 Transferable utilities | 34

4 4.1 4.2 4.3 4.4 4.5 4.6 4.6.1 4.6.2 4.7

Monge partitions | 39 Capacities | 39 First paradigm: the big brother | 41 Second paradigm: free market | 42 The big brother meets the free market | 43 All the ways lead to stable subpartitions | 46 Weak definition of partitions | 47 Kantorovich relaxation of (sub)partitions | 49 Birkhoff theorem | 51 Summery and beyond | 53

VIII | Contents

Part II: Multipartitions 5 5.1 5.2 5.2.1 5.2.2 5.3 5.3.1

Weak multipartitions | 57 Multipartitions | 57 Feasibility conditions | 59 Dual representation of weak (sub)partitions | 59 Proof of Theorem 5.1 | 61 Dominance | 64 Minimal elements | 68

6 6.1 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.3

Strong multipartitions | 71 Strong partitions as extreme points | 71 Structure of the feasibility sets | 72 Coalitions and cartels | 73 Fixed exchange ratio | 75 Proofs | 76 An application: two states for two nations | 80 Further comments | 81

7 7.1 7.1.1 7.1.2 7.1.3 7.2 7.2.1 7.2.2 7.2.3 7.2.4

Optimal multipartitions | 83 Optimality within the weak partitions | 83 Extension to hyperplane | 83 Optimal multi(sub)partitions: extended setting | 87 Price adaptation and escalation | 91 Optimal strong multipartitions | 94 Example of escalation | 95 Uniqueness for a prescribed capacity | 96 Uniqueness within the feasibility domain | 99 The MinMax theorem: a unified formulation | 100

8 8.1 8.2 8.2.1 8.2.2

Applications to learning theory | 103 Maximal likelihood of a classifier | 103 Information bottleneck | 106 Minimizing the distortion | 108 The information bottleneck in the dual space | 110

Part III: From optimal partition to optimal transport and back 9 9.1 9.2

Optimal transport for scalar measures | 117 General setting | 117 Duality | 118

Contents | IX

9.3 9.3.1 9.4 9.4.1 9.4.2

Deterministic transport | 121 Solvability of the dual problem | 123 Metrics on the set of probability measures | 125 Special cases | 126 McCann interpolation | 128

10 Interpolated costs | 131 10.1 Introduction | 131 10.1.1 Semi-finite approximation: the middle way | 132 10.2 Optimal congruent partitions | 133 10.3 Strong partitions | 136 10.4 Pricing in hedonic market | 138 10.5 Dependence on the sampling set | 139 10.5.1 Monotone improvement | 139 10.5.2 Asymptotic estimates | 140 10.6 Symmetric transport and congruent multi-partitions | 143

Part IV: Cooperative and non-cooperative partitions 11 Back to Monge: individual values | 149 11.1 The individual surplus values | 154 11.2 Will wiser experts always get higher values? | 157 11.2.1 Proofs | 160 12 Sharing the individual value | 167 12.1 Maximizing the agent’s profit | 169 12.2 Several agents: Nash equilibrium | 170 12.3 Existence of Nash equilibrium | 171 12.4 Efficiency | 171 12.4.1 Efficiency for agents of comparable utilities | 173 12.4.2 Efficiency under commission strategy | 174 12.5 Free price strategy | 175 12.5.1 Where Nash equilibrium meets efficiency | 176 13 Cooperative partitions | 177 13.1 Free price strategy | 177 13.2 Cooperative games – a crash review | 178 13.2.1 Convex games | 180 13.3 Back to cooperative partition games | 182 13.3.1 Flat prices strategy: regulation by capacity | 182 13.3.2 Coalition games under comparable utilities | 185

X | Contents A A.1 A.2 A.3 A.4 A.5 A.6

Convexity | 189 Convex sets | 189 Convex functions | 190 Lower semi-continuity | 192 Legendre transformation | 193 Subgradients | 194 Support functions | 195

B B.1 B.2 B.3

Convergence of measures | 197 Total variation | 197 Strong convergence | 198 Weak* convergence | 199

Bibliography | 203 Index | 207

How to read this book? The introduction (Chapter 1) provides an overview on the content of the book. Chapters 2 and 3 are independent of the rest of this book. Other than that, Chapter 4 is the core of this book, and it is a prerequisite for the subsequent chapters. The readers who are mainly interested in the applications to economics and game theory may jump from Chapter 4 to Part IV, starting from Section 11.1 and taking Theorem 11.1 for granted, and also Section 10.4. Some of these readers may find also an interest in Chapter 6, which, unfortunately, is not independent of Chapter 5, The reader interested in application to learning theory may skip from Chapter 4 to Section 8.1, but it is recommended to read Part II (or, at least go over the definitions in Chapter 5) before reading Section 8.2 on the information bottleneck. It is also possible to read Part III after Chapter 4 which, except Section 10.6, is independent of the rest.

https://doi.org/10.1515/9783110635485-202

Notations The following is a (non-exhaustive) list of notations used throughout the book. Other notations will be presented at the first time used. 1.

ℝ is the field of real numbers, ℝ+ the real non-negative numbers, ℝ++ the real positive numbers, ℝ− the real non-positive numbers, and ℝ−− the real negative numbers. 2. For x, y ∈ ℝ, min(x, y) := x ∧ y, max(x, y) := x ∨ y. 3. ΔN (γ) := {(x1 , . . . , xN ) ∈ ℝN+ , ∑Ni=1 xi = γ}. 4. ΔN (γ) := {(x1 , . . . , xN ) ∈ ℝN+ , ∑Ni=1 xi ≤ γ}. 5. 𝕄+ (N, J) := ℝN+ ⊗ ℝJ+ . It is the set of N × J matrices of non-negative real numbers. Likewise, 𝕄󸀠 (N, J) := ℝN ⊗ ℝJ is the set of N × J matrices of real numbers. 6. For M⃗ = {mi,j } ∈ 𝕄+ (N, J) and P⃗ = {pi,j } ∈ 𝕄󸀠 (N, J), P⃗ : M⃗ := tr(M⃗ P⃗ t ) = ∑Ni=1 ∑Jj=1 pi,j mi,j . 7. (X, ℬ) is a compact measure space, and ℬ is the Borel σ-algebra on X. 8. ℳ(X) is the set of Borel measures on X. ℳ+ ⊂ ℳ is the set of non-negative measures, and ℳ1 are the probability measures, namely μ(X) = 1.

https://doi.org/10.1515/9783110635485-203

1 Introduction 1.1 The fully discrete case1 Imagine a set ℐm composed of N men and a set ℐw composed of N women. Your task is to form N married pairs {ii󸀠 } ⊂ ℐm × ℐw out of these sets, where each pair is composed of a single man i ∈ ℐm and a single woman i󸀠 ∈ ℐw , and make everybody happy. This is the celebrated stable marriage problem. What is the meaning of “making everybody happy”? There is, indeed, a very natural definition for it, starting from the definition of a blocking pair. A blocking pair is an unmarried couple (a man and a woman) who prefer each other over their assigned spouses. The existence of a blocking pair will cause two couples to divorce and may start an avalanche destabilizing all the assigned matchings. The definition of a stable marriage (which, in our case, is a synonym to “happy marriage”) is There are no blocking pairs.

The main focus in this book is on the transferable model, which assumes a somewhat materialistic point of view. A married couple ii 󸀠 ∈ ℐm × ℐw can share a reward θ(i, i 󸀠 ) ≥ 0 (say, in US dollars).

Suppose now that you assigned man i to woman i󸀠 and man j ≠ i to woman j󸀠 ≠ i󸀠 . A necessary condition for a stable marriage is θ(i, i󸀠 ) + θ(j, j󸀠 ) ≥ θ(i, j󸀠 ) + θ(j, i󸀠 ).

(1.1)

Indeed, assume the couple ii󸀠 splits the reward between themselves, so that i cuts ui = αθ(i, i󸀠 ) dollars while i󸀠 cuts vi󸀠 = (1 − α)θ(i, i󸀠 ) dollars, where α ∈ (0, 1). Likewise, the couple jj󸀠 splits their reward according to the cuts uj = βθ(j, j󸀠 ) and vj󸀠 = (1 − β)θ(j, j󸀠 ), where β ∈ (0, 1). If θ(i, i󸀠 ) + θ(j, j󸀠 ) < θ(i, j󸀠 ) + θ(j, i󸀠 ), then θ(i, j󸀠 ) + θ(j, i󸀠 ) > ui + vi󸀠 + uj + vj󸀠 , so either θ(i, j󸀠 ) > ui + vj󸀠 or θ(j, i󸀠 ) > uj + vi󸀠 (or both). In any case at least one of the new pairs ij󸀠 , ji󸀠 can share a reward bigger than the one they could get from their former matching, and thus improve their individual cuts. Hence at least one of the pairs ij󸀠 or ji󸀠 is a blocking pair. From the above argument we conclude that (1.1) for any two matched pairs is a necessary condition for the marriage to be stable. Is it also sufficient? 1 Part of this chapter was published by the author in [52]. https://doi.org/10.1515/9783110635485-001

2 | 1 Introduction Suppose the pairs (i1 i1󸀠 ), . . . , (ik ik󸀠 ), k ≥ 2, are matched. The sum of the rewards for these couples is ∑kl=1 θ(il , il󸀠 ). Suppose they perform a “chain deal” such that man il 󸀠 marries woman il+1 for 1 ≤ l ≤ k − 1, and the last man ik marries the first woman i1󸀠 . The 󸀠 󸀠 net reward for the new matching is ∑k−1 l=1 θ(il , il+1 ) + θ(ik , i1 ). A similar argument implies that a necessary condition for a stable marriage is that this new reward will not exceed the original net reward for this matching, that is, For any choice of matched pairs i1 i1󸀠 , . . . , ik ik󸀠 , k

󸀠 ∑ θ(il , il󸀠 ) − θ(il , il+1 )≥0 l=1

(1.2)

󸀠 (where ik+1 := i1󸀠 ).

Condition (1.2) generalizes (1.1) to the case k ≥ 2. It is called cyclical monotonicity. It is remarkable that cyclical monotonicity is, indeed, equivalent to the stability of matching {ii󸀠 } (i. e., to the absence of blocking pairs). From cyclical monotonicity we can conclude directly an optimality characterization of stable matching. In fact, this is an equivalent definition of stable marriage in the transferable case: The marriage {ii 󸀠 } is stable if and only if it maximizes the total reward among all possible 1 − 1 matchings i ∈ ℐm → τ(i) ∈ ℐw , that is, N

N

i=1

i=1

∑ θ(i, i 󸀠 ) ≥ ∑ θ(i, τ(i)).

(1.3)

Another very important notion for the marriage problem (and, in general, for any cooperative game) is the notion of feasibility set and core. The feasibility set is the collection of men’s cuts ui and women’s cuts vj󸀠 which satisfy the feasibility condition: ui + vj󸀠 ≥ θ(i, j󸀠 )

(1.4)

for all ij󸀠 ∈ ℐm × ℐw . The core of a given matching {ii󸀠 } is composed of all such cuts (u1 , . . . , uN ; v1 , . . . , vN ) in the feasibility set which satisfies the equality ui + vi󸀠 = θ(i, i󸀠 ) for any matched pair ii󸀠 . The matching {ii 󸀠 } ⊂ {ℐm × ℐw } is stable if and only if the associated core is not empty.

There is another, dual optimality formulation for a stable matching via the feasibility set:

1.1 The fully discrete case

| 3

The cuts u01 , . . . , u0N ; v10 , . . . ,vN0 form a core if and only if they are a minimizer of the total cut ∑N1 ui + vi within the feasibility set (1.4): N

N

1

1

∑ u0i + vi0 ≤ ∑ ui + vi .

(1.5)

In particular, if u01 , . . . , vN0 is such a minimizer, then for any man i ∈ ℐm there exists at least one woman i 󸀠 ∈ ℐw and for any woman i 󸀠 ∈ ℐw there exists at least one man i ∈ ℐm for which the equality u0i + vi0󸀠 = θii󸀠 holds, and the matching {ii 󸀠 } ⊂ ℐm × ℐw is stable.

Each of the two dual optimality characterizations (1.3), (1.5) of stable matching guarantees that for any choice of the rewards {θ(i, j)}, a stable matching always exists. There are other ways to define a blocking pair. A natural way is the non-transferable marriage. In the non-transferable marriage game each man and woman have a preference list, by which he/she rates the women/men in the group. This is the celebrated marriage problem of Gale and Shapley (who won a Nobel Prize in economics in 2012). We may quantify the Gale–Shapley game (after all, we live in a materialistic world). Assume a paring of man i and woman j󸀠 will guarantee a cut θm (i, j󸀠 ) to the man and θw (i, j󸀠 ) to the woman. This will induce the preference list for both men and women: Indeed, the man i will prefer woman i󸀠 over j󸀠 if and only if θm (i, i󸀠 ) > θm (i, j󸀠 ). Likewise, woman i󸀠 will prefer man i over j if and only if θw (i, i󸀠 ) > θw (j, i󸀠 ). A blocking pair for a matching {ii󸀠 } ⊂ ℐm × ℐw is, then, a pair ij󸀠 such that j󸀠 ≠ i󸀠 and both θm (i, j󸀠 ) ≥ θm (i, i󸀠 )

and θw (i, j󸀠 ) ≥ θw (j, j󸀠 )

are satisfied (where at least one of the inequalities is strong). Gale–Shapley-stability for a set of rewards {θm , θw } does not imply the stability of the transferable game where θ = θm +θw , where each couple is permitted to share their individual rewards (and neither the opposite). A simple example (N = 2): θm m1 m2

w1 1 0

w2 0 ; 1

θw m1 m2

w1 1 0

w2 5 . 1

The matching {11, 22} is Gale–Shapley-stable. Indeed θm (1, 1) = 1 > θm (1, 2) = 0 while θm (2, 2) = 1 > θm (2, 1) = 0, so both men are happy, and this is enough for Gale–Shapley-stability, since neither {12} nor {21} is a blocking pair. On the other hand, if the married pairs share their rewards θ(i, j 󸀠 ) = θm (i, j 󸀠 ) + θw (i, j 󸀠 ), we get θ m1 m2 so

w1 2 0

w2 5 2

θ(1, 1) + θ(2, 2) = 4 < 5 = θ(1, 2) + θ(2, 1),

thus {21, 12} is the stable marriage in the transferable setting.

4 | 1 Introduction On top of it, there exists a whole world of marriage games which contain the transferable and Gale–Shapley games as special cases. There is a deep theorem which guarantees the existence of a stable marriage for a wide class of partially transferable games, starting from the fully transferable, all the way to Gale–Shapley. The proof of this theorem is much simpler in the transferable case (due to the optimality characterization) and the Gale–Shapley case (due to the celebrated Gale–Shapley algorithm, which is described in Section 2.1). However, there is an essential difference between the transferable game and all other cases. To the best of our knowledge: The transferable marriage game is the only one which is variational, i. e., whose stable solutions are characterized by an optimality condition.

A discussion on the marriage problem and some of its generalizations is given in Chapter 2.

1.2 Many to few: partitions We may extend the marriage paradigm to a setting of matching between two sets of different cardinality. Suppose ℐ = {1, . . . , N} is a set representing experts (or sellers) and X is a much larger (possibly infinite) set representing, say, the geographical space in which the customers (or consumers) live. We consider (X, ℬ, μ) as a measure space, equipped with a σ-algebra ℬ and a positive measure μ. We shall always assume that X is also a compact space and ℬ a Borel. In the expert–customer interpretation μ(B) it is the number of customers living in B ∈ ℬ. We also associate any i ∈ ℐ with a capacity mi > 0. This can be understood as the maximal possible number of customers the expert i can serve. A measurable matching τ : X → ℐ can be represented by a partition A⃗ = (A0 , A1 , . . . , AN ), where Ai = τ−1 ({i}) ∈ ℬ, i ∈ ℐ ∪ {0} are pairwise disjoint. The set Ai , i ∈ ℐ represents the geographical domain in X served by expert i, and μ(Ai ) represents the number of customers served by i. The set A0 represents a domain which is not served by any of the experts. A feasible partition must satisfy the constraint2 μ(Ai ) ≤ mi ,

i ∈ ℐ.

(1.6)

Let us consider the generalization of the transferable marriage game in this context. The utility of the assignment of x ∈ X to i ∈ ℐ is given by the function θ ∈ C(X × ℐ ). This function is assumed to be non-negative. We usually denote θ(x, i) := θi (x) for i ∈ ℐ , x ∈ X and θ0 (x) ≡ 0 is the utility of non-consumer. The optimal partition A01 , . . . , A0N is the one which realizes the maximum 2 See Section 1.3.1 below for a discussion in the case of inequality (1.6) vs equality.

1.2 Many to few: partitions | 5

N

N

∑ ∫ θi (x)dμ ≥ ∑ ∫ θi (x)dμ i=1 0 Ai

i=1 A

(1.7)

i

for any feasible subpartition A1 , . . . , AN verifying (1.6).

The assumption Ai ⊂ X seems to be too restrictive. Indeed, an expert can serve only part of the customers at a given location. So, me may extend the notion of partition to a weak partition. A weak partition is represented by N non-negative measures μi on (X, ℬ) verifying the constraints μi ≥ 0,

N

∑ μi ≤ μ, i

μi (X) ≤ mi .

(1.8)

Of course, any strong partition A1 , . . . , AN is a weak partition, where μi = μ⌊Ai (the restriction of μ to Ai ). The general notion of stable marriage in the fully discrete case (ℐm , ℐw ) can be generalized to stable partition in the semi-discrete case (X, ℐ ). A natural generalization of (1.6) leads to a stable weak partition μ⃗ 0 := (μ01 , . . . , μ0N ) obtained by maximizing the total utility N

N

∑ ∫ θi (x)dμ0i ≥ ∑ ∫ θi (x)dμi i=1 X

for any feasible subpartition verifying (1.8).

i=1 X

(1.9)

As in the fully discrete setting of the marriage problem, we may consider other, nontransferable partitions. In particular, the Gale–Shapley marriage game is generalized as follows. Assume that ℐ stands for a finite number of firms and X for the set of potential employees. Let e(x, i) be the reward for x if hired by i, and f (x, i) the reward of firm i employing x. The condition for a strong, stable partition A1 , . . . , AN under nontransferable assumption subjected to the capacity constraint μ(Ai ) ≤ mi is: Either e(x, i) ≥ e(x, j) for x ∈ Ai , j ∈ ℐ or there exists y ∈ Aj , j ≠ i, where f (y, j) > f (x, j).

In Chapter 3 we consider the partition problem for both the completely transferable and non-transferable cases. In Chapter 4, as well as in the rest of the book, we restrict ourselves to the fully transferable case. There we lay the foundations of duality theory for optimal partitions. In the case of equality in (1.6), (1.8)3 and ∑ mi = μ(X), this dual formulation takes the 3 See Section 1.3.1 below.

6 | 1 Introduction form of minimizing the convex function N

(p1 , . . . , pN ) ∈ ℝN 󳨃→ Ξ(p1 , . . . , pN ) + ∑ pi mi ∈ ℝ, i=1

(1.10)

where Ξ(p1 , . . . , pN ) := ∫ max (θi (x) − pi ) dμ. X

1≤i≤N

In the agent–customer interpretation, the optimal pi stands for the equilibrium price charged by agent i for her service. The inequality N

M

Ξ(p1 , . . . , pN ) + ∑ pi mi ≥ ∑ ∫ θi (x)dμ i=1

i=1 A

(1.11)

i

plays a fundamental role in Part II.

1.3 Optimal transport in a nutshell Both the transferable marriage and partition problems are special cases of the Monge problem in optimal transport. The original formulation of the Monge problem is very intuitive. It can be stated as follows: Given a pile of sand X and a container Y of the same volume, what is the best plan of moving the sand from the pile in order to fill the container?

What do we mean by “a plan”? Let μ ∈ ℳ+ (X) be a measure on X signifying the distribution of sand. Let ν ∈ ℳ+ (Y) be a measure on Y signifying the distribution of free space in the container. The balanced condition, representing the statement “same volume” above, takes the form μ(X) = ν(Y).

(1.12)

A strong plan is a mapping T : X → Y which transports the measure μ to ν, that is, T# μ = ν, namely, μ(T −1 (B)) = ν(B) for every measurable set B ⊂ Y.

(1.13)

1.3 Optimal transport in a nutshell | 7

The “best plan” is the one which minimizes the average distance ∫ d(x, T(x))μ(dx) X

where d(⋅, ⋅) is a metric in the space, among all plans. The interest of Monge was mainly geometrical. In his only (known) paper on this subject [35] he discovered some fundamental properties of the minimizer and connected the notion of transport rays and wavefronts in optics to this geometrical problem. In the generalized version of the Monge problem the distance function (x, y) 󳨃→ d(x, y), x, y ∈ X, is replaced by a cost of transportation (x, y) 󳨃→ c(x, y), where x ∈ X, y ∈ Y. In particular, X and Y can be different domains. The Monge problem takes the form of the minimization problem c(μ, ν) := min ∫ c(x, T(x))μ(dx) T

(1.14)

X

among all maps transporting the probability measure μ on X to ν on Y (i. e., T# μ = ν). In the context of expert–customer (which we adopt throughout most of this book), it is more natural to replace the cost c by the utility θ, which we want to maximize. Evidently, one may switch from c(x, y) to θ(x, y) = −c(x, y) and from (1.14) to θ(μ, ν) := max ∫ θ(x, T (x))μ(dx). T# μ=ν

(1.15)

X

After this pioneering publication of Monge, the problem fell asleep for about 160 years, until Kantorovich’s paper in 1941 [28]. Kantorovich’s fundamental observation was that this problem is closely related to a relaxed problem on the set of two-point probability measures π = π(dxdy) θ(μ, ν) := max ∫ ∫ θ(x, y)π(dxdy), π∈Π(μ,ν)

(1.16)

X Y

where Π(μ, ν) is the set of “weak plans” composed of point distributions on X×Y whose marginals are μ and ν: Π(μ, ν) := {π ∈ ℳ+ (X × Y); ∫ π(dxdy) = ν(dy), ∫ π(dxdy) = μ(dx)}. X

(1.17)

Y

The optimal measure π(A × B) represents the probability of transporting goods located in the measurable set A ⊂ X to B ⊂ Y. The disintegration π(A, B) = ∫ Px (B)μ(dx) A

(1.18)

8 | 1 Introduction reveals the conditional probability Px of the transportation from x ∈ X to B ⊂ Y. Thus, we can interpret Kantorovich’s transport plan as a stochastic transport. In contrast, deterministic transport T via Monge’s paradigm is the special case where the conditional probability Px takes the form Px (dy) = δy−T(x) . The transferable marriage problem is a simplified version of an optimal transport plan. Here we replaced the atoms x ∈ X and y ∈ Y by finite, discrete sets of men i ∈ ℐm and women i󸀠 ∈ ℐw of the same cardinality N. The measures μ, ν are just the uniform discrete measures μ({i}) = ν({j󸀠 }) = 1 for all i ∈ ℐm and j󸀠 ∈ ℐw , while the utility θ(x, y) is now represented by an N × N matrix θ(i, j󸀠 ). The Monge plan verifying (1.16) now takes the form of the assignment given in terms of a permutation i󸀠 = τ(i) which maximizes the total reward of matching N

τ ⇒ ∑ θ(i, τ(i)).

(1.19)

i=1

The Kantorovich program replaces the deterministic assignment by a probabilistic j󸀠 one: πi := π({i}, {j󸀠 }) ≥ 0, is the probability of assigning i to j󸀠 . The optimal solution is then reduced to the linear programming of maximizing N

N

j󸀠

∑ ∑ πi θ(i, j󸀠 )

(1.20)

i=1 j󸀠 =1 j󸀠

over all stochastic N × N matrices {πi }, i. e., these matrices which satisfy the 3N linear constraints N

j󸀠

N

j󸀠

∑ πi = ∑ πi = 1, i=1

j󸀠 =1

j󸀠

πi ≥ 0.

The Birkhoff theorem4 assures us that the optimal solution of this stochastic assignment problem (1.20) is identical to the solution of the deterministic version (1.19). In j particular, the optimal stochastic matrix {πi } is a permutation matrix δτ(i)−j associated with the permutation τ. Likewise, the transferable partition in the balanced case ∑i∈ℐ mi = μ(X) corresponds to a solution of the Kantorovich problem where the target space Y is given by the discrete space ℐ of finite cardinality N. The measure ν is given by the capacities mi := ν({i}). The utility θ(x, y) is represented by θi (x), where i ∈ ℐ . A strong partition in X corresponds to a transport (1.13), where Ai = T −1 ({i}). The optimal partition (1.7) corresponds to the solution of the Monge problem (1.15). 4 See Section 4.6.2.

1.3 Optimal transport in a nutshell | 9

The weak optimal partition (1.9) is nothing but the Kantorovich relaxation (1.16) to the deterministic transport partition problem. Indeed, the set Π(μ, ν) (1.17) is now reduced to the set of all weak partitions π(dx × {i}) := μi (dx) via the weak partition N

⃗ := {μ⃗ := (μ1 , . . . , μN ), μi (X) = mi , ∑ μi = μ}. Π(μ, m) 1

As a particular example we may assume that X = {x1 , . . . , xN ∗ } is a discrete case as well. In that case we denote θi (xj ) := θ(i, j), μ({xi }) := m∗i . In the balanced case

∑Ni=1 mi = ∑Ni=1 m∗i we get the optimal weak partition μ0i ({xj }) := (πi0,1 , . . . , πi0,N ) as ∗

∗

N N∗

0.j

j

{πi } = arg max ∑ ∑ πi θ(i, j), {mji } i=1 j=1

j

where {πi } verifying (1.8) in the case of equality j

πi ≥ 0;

N∗

j

∑ πi = m∗i , j=1

N

j

∑ πi = mj ; i=1

(i, xj ) ∈ ℐ × X.

(1.21)

We may recover the fully discrete transferable marriage (1.19) in the particular case N ∗ = N and mi = m∗j = 1 for 1 ≤ i, j ≤ N. The Birkhoff theorem hints that the case where the optimal partition μ⃗ 0 in (1.9) is a strong subpartition μ0i = μ⌊A0i is not so special, after all... 1.3.1 Unbalanced transport The case of unbalanced transport μ(X) ≠ ν(Y) deserves special attention. Note, in particular, that in (1.6) we used the inequality μ(Ai ) ≤ mi . If the utilities θi are non-negative and if ∑N1 mi ≤ μ(X), then it is evident that the optimal partition will satisfy the equality μ(A0i ) = mi (same for (1.8), (1.9)). This presents no conceptual new case, since we can define m0 := μ(X) − ∑N1 mi and A0 := X − ⋃N1 Ai constrained by μ(A0 ) = m0 , representing the non-consumers in the populations. This reduces the problem to the case of the equality ∑N0 mi = μ(X), where the utility of non-consuming is θo ≡ 0. In the dual formulation we may assign, in the case m0 > 0, the price p0 = 0 for non-consuming. In that case, inequality (1.11) will be preserved, where we integrate only on the positive part of θi − pi , i. e., (θi (x) − pi )+ := (θi (x) − pi ) ∨ 0. Thus, Ξ is replaced by Ξ+ (p1 , . . . , pN ) := ∫ max (θi (x) − pi )+ dμ. X

1≤i≤N

10 | 1 Introduction In the same way we may adopt in the Monge problem (1.15) the case μ(X) > ν(Y) by adding an auxiliary point y0 to Y and extend ν to Y ∪{y0 } such that ν({y0 }) = μ(X)−ν(Y), together with θ(x, y0 ) = 0 for any x ∈ X. The case μ(X) < ν(Y) is treated similarly. We just add a virtual point x0 to X, assign μ({x0 }) = ν(Y) − μ(X) and θ(x0 , y) = 0 for any v ∈ Y. In the semi-discrete case ∑N1 mi > μ(X) this changes (1.11) into N

M

Ξ(p1 , . . . , pN ) + ∑ pi mi − m0 min pi ≥ ∑ ∫ θi (x)dμ, i

i=1

i=1 A

i

where m0 := ∑N1 mi − μ(X) in that case.

1.4 Vector-valued transport and multipartitions A natural generalization of the optimal transport is optimal vector-valued transport. Here we replace the measures μ, ν by ℝJ+ -valued measures μ̄ := (μ(1) , . . . , μ(J) ) ∈ ℳJ+ (X),

ν̄ := (ν(1) , . . . , ν(J) ) ∈ ℳJ+ (Y),

and we denote μ = |μ|̄ := ∑J1 μ(j) , ν = |ν|̄ := ∑J1 ν(j) . The set Π(μ, ν) (1.17) is generalized into Π(μ,̄ ν)̄ := {π ∈ ℳ+ (X × Y);

∫ X

dμ(j) (x)π(dxdy) = ν(j) (dy)}, dμ

(1.22)

where dμi /dμ, dνi /dν stands for the Radon–Nikodym derivative. In general the set Π(̄ μ,̄ ν)̄ can be an empty one. If Π(μ,̄ ν)̄ ≠ 0, then we say that μ̄ dominates ν.̄ This is an order relation (in particular transitive), denoted by μ̄ ≻ ν.̄

(1.23)

The generalization of the Kantorovich problem (1.16) takes the form θ(μ,̄ ν)̄ := max ∫ ∫ θ(x, y)π(dxdy), π∈Π(μ,̄ ν)̄

θ(μ,̄ ν)̄ = ∞ if μ̄ ⊁ ν.̄

X Y

Several recent publications deal with a notion of vector-valued (or even matrix-valued) optimal transport. See, in particular, [53] as well as related works [12, 23, 45, 10, 37, 11]. There is, however, a fundamental difference between our notion of vector transport and those publications, since (1.22) implies a single transport plan for all components of the vector.

1.4 Vector-valued transport and multipartitions | 11

A possible motivation for studying such a transport concerns some application to learning theory. A vector-valued measure μ̄ := (μ(1) , . . . , μ(J) ) on a set X is interpreted as a distribution of a classifier of a label j ∈ {1, . . . , J} given a sample x in some feature space X. The object of learning is to model this classifier by a simpler one on a finite sample space ℐ , while preserving as much as possible the information stored in the given classifier. This subject is discussed in Chapter 8. In Part II we consider an implementation of ℝJ+ -valued transport to multipartitions. Here we replace the space Y with the discrete space ℐ = {1, . . . , N}, and the (j) (j) ℝJ -valued measure ν̄ is represented by an N × J matrix M⃗ := {mi }, where mi stands (j) for ν ({i}). A multipartition of X subjected to M⃗ is a partition of X into mutually disjoint measurable sets A1 , . . . , AN ⊂ X satisfying for i = 1, . . . , N, j = 1 . . . J, ∪Ai = X.

μ(j) (Ai ) = mi , (j)

(1.24)

Similarly, a weak multipartition stands for N non-negative measures μ1 , . . . , μN verifying N

J

1

j=1

∑ μi = μ := ∑ μ(j) .

(1.25)

The induced weak partition μ̄ i := (μ(1) , . . . , μ(J) ), i = 1, . . . , N, is defined by i i μi (dx) := (j)

dμi (x)μ(j) (dx). dμ

Such a weak partition is assumed to satisfy μi (X) = mi , (j)

(j)

i = 1 . . . N, j = 1 . . . J.

(1.26)

An optimal multipartition μ⃗ 0 := (μ01 , . . . , μ0N ) is a natural generalization of (1.8): It is the one which maximizes N

N

⃗ := ∑ ∫ θi (x)dμ0 ≡ max ∑ ∫ θi (x)dμi θ(μ;̄ M) i i=1 X

μ⃗

i=1 X

(1.27)

(j) among all weak partitions μ⃗ = (μ1 , . . . , μN ) verifying (1.26) for the assigned μ,̄ M⃗ := {mi }.

At the first step, we should ask ourselves if such a weak multipartition exists at all. By (1.25) we can see that a necessary condition for this is the componentwise balance (j) ∑Ni=1 mi = μ(j) (X) for 1 ≤ j ≤ J. In general, however, this is not a sufficient condition. ⃗ we say that μ̄ dominates M⃗ If a weak partition verifying (1.26) exists for a pair μ,̄ M, ⃗ ⃗ ̄ and denote it by μ ≻ M. The set of all N × J matrices M satisfying μ̄ ≻ M⃗ is denoted by ̄ The connection with (1.23) is the following. ̄ We denote μ ≻N ν if ΔN (μ)̄ ⊃ ΔN (ν). ΔN (μ).

12 | 1 Introduction

Theorem. We have μ̄ ≻ ν̄ if and only if μ̄ ≻N ν̄ for any N ∈ ℕ.

The feasibility condition for (1.27), namely, the condition ΔN (μ)̄ ≠ 0, and the characterization of ΔN (μ)̄ in general are addressed in Chapter 5. The function Ξ0 (P;⃗ μ)̄ := ∫ max p⃗ i ⋅ X

1≤i≤N

dμ̄ dμ dμ

J

plays a central role. Here p⃗ ∈ ℝ and P⃗ := (p⃗ 1 , . . . , p⃗ N ) ∈ ℝN×J is an N × J matrix. The main result of this chapter is the following: The set ΔN (μ)̄ is closed and convex. We have M⃗ ∈ ΔN (μ)̄ (i. e., μ̄ ≻ M)⃗ if and only if one of the following equivalent conditions holds: – Ξ0 (P;⃗ μ)̄ − P⃗ : M⃗ ≥ 0, where M⃗ := (m̄ 1 , . . . , m̄ N ), m̄ i := (m(1) , . . . , m(J) ), and P⃗ : M⃗ := ∑N p⃗ i ⋅ m̄ i ; –

i

for any convex function f : ℝN → ℝ, ∫f ( X

i=1

i

N d μ̄ m̄ ) dμ ≥ ∑ |m̄ i |f ( i ) , dμ | m̄ i | i=1

where |m̄ i | = ∑Jj=1 mi . (j)

The existence of strong partitions verifying (1.24) is discussed in Chapter 6. In particular, we obtain the following: If μ̄ ≻ M⃗ and

̄ μ (x ∈ X ; p⃗ ⋅ d μ/dμ = 0) = 0

for any p⃗ ∈ ℝJ , p⃗ ≠ 0,

(1.28)

then there exists a strong partition A1 , . . . , AN verifying (1.24) corresponding to M.⃗ Moreover, if there exists P⃗ 0 := (p⃗ 01 , . . . , p⃗ 0N ) ≠ 0 which verifies Ξ0 (P⃗ 0 ; μ)̄ − P⃗ 0 : M⃗ = 0 and satisfies p⃗ 0i ≠ p⃗ 0j for i ≠ j, then the strong partition is unique.

More generally, if ℐ is decomposed into k disjoint subsets ℐ1 , . . . , ℐk and p⃗ 0i ≠ p⃗ 0j if i ∈ ℐm , j ∈ ℐn , and n ≠ m, then there exists a unique k-partition 𝒜1 , . . . , 𝒜k of X such that any partition verifying (1.24) corresponding to M⃗ satisfies Ai ⊂ 𝒜m

if and only if i ∈ ℐm .

In Chapter 7 we consider the optimization problem for multipartitions. The function dμ̄ Ξθ (P;⃗ μ)̄ := ∫ max [θi (x) + pi ⋅ ] dμ 1≤i≤N dμ + X

plays a central role for the optimization. One of the main results of this chapter is the following:

1.5 Cooperative and non-cooperative partitions |

13

If μ̄ ≻ M,⃗ then the optimal transport (1.27) is given by ⃗ = inf Ξθ (P;⃗ μ)̄ − P⃗ : M.⃗ θ(μ,̄ M) P⃗

(1.29)

Moreover, if (1.28) holds, then there is a strong partition which verifies (1.27).

1.5 Cooperative and non-cooperative partitions In Part IV we return to the scalar transport case J = 1 and discuss partitions under both cooperation and competition of the agents. Taking advantage of the uniqueness result for partition obtained in Chapter 7.2, we define, in Chapter 13.3, the individual value Vi of an agent i as the surplus value she creates for her customers: Vi := ∫ θi (x)dμ, Ai

where Ai is the set of the customers of i under the optimality condition. We address the following question: What is the effect of increase of the utility θi (x) of agent i on its individual value Vi , assuming the utilities of the other agents as well as the capacities mk = μ(Ak ) are preserved for all agents?

The answer to this question is somewhat surprising. It turns out that the individual value may decrease in that case. In Theorems 11.2–11.4 we establish sharp quantitative estimates of the change of the individual value. In Chapter 12 we deal with different possibilities of sharing the individual value Vi produced by the agent i with her customers Ai . The most natural strategy is “flat price,” where agent i charges a constant price pi for all her customers, so her profit is Pi := pi μ(Ai ). Since μ(Ai ) is determined by the prices p1 , . . . , pN imposed by all other agents, we obtain a competitive game where each agent wishes to maximize her profit. This leads us naturally to the notion of Nash equilibrium. We also discuss other strategies, such as commission, where agent i charges a certain portion qi θi , where qi ∈ (0, 1), and hence Pi = qi Vi . Motivated by these results we ask the natural question regarding cooperation of agents: Suppose a subgroup of agents 𝒥 ⊂ ℐ := {1, . . . , N} decide to form a coalition (cartel), such that the utility of this coalition is the maximum of utilities of its agents, θ𝒥 (x) := maxj∈𝒥 θj (x), and the capacity is the sum of the capacities m𝒥 := ∑i∈𝒥 mi . The stability of the grand coalition θℐ = maxj∈ℐ θj and mℐ = μ(X) is addressed in Chapter 13. This leads us to discuss cooperative games for transferable utilities. In some special cases we establish the stability of the grand coalition 𝒥 = ℐ .

|

Part I: Stable marriage and optimal partitions

2 The stable marriage problem Obviously, marriage is not a synonym for morality. But stable marriages and families do encourage moral behavior. (Gary Bauer)

2.1 Marriage without sharing Consider two sets of N elements each: A set of men (ℐm ) and a set of women (ℐw ). Each man in i ∈ ℐm lists the women according to his own preference: For any j1 , j2 ∈ ℐw j1 ≻i j2 iff i prefers j1 over j2 .

(2.1)

Likewise, each woman j ∈ ℐw lists the men in ℐm according to her preference: For any i1 , i2 ∈ ℐm i1 ≻j i2 iff j prefers i1 over i2 .

(2.2)

Here ≻i , ≻j are complete order relations, namely: 1. for any i ∈ ℐm and j1 ≠ j2 ∈ ℐw either j1 ≻i j2 or j2 ≻i j1 (but not both); 2. j1 ≻i j2 , j2 ≻i j3 implies j1 ≻i j3 for any distinct triple j1 , j2 , j3 ∈ ℐw ; 3. likewise for ≻j where j ∈ ℐw . A matching τ is a bijection i ↔ τ(i): Any man i ∈ ℐm marries a single woman τ(i) ∈ ℐw , and any woman j ∈ ℐw is married to a single man τ−1 (j) ∈ ℐm . A blocking pair (i, j) ∈ ℐm × ℐw is defined as follows: – j and i are not married (j ≠ τ(i)); – i prefers j over his mate τ(i): j ≻i τ(i); – j prefer i over her mate τ−1 (j): i ≻j τ−1 (j). Definition 2.1.1. A marriage τ is called stable if and only if there are no blocking pairs. This is a very natural (although somewhat conservative) definition of stability, as the existence of a blocking pair will break two married couples and may disturb the happiness of the rest. The question of existence of a stable marriage is not trivial. It follows from a celebrated, constructive algorithm due to Gale and Shapley [17], which we describe below.

https://doi.org/10.1515/9783110635485-002

18 | 2 The stable marriage problem 2.1.1 Gale–Shapley algorithm Freedom’s just another word for nothin’ left to lose. (Jenis Joplin)

1.

At the first stage, each man i ∈ ℐm proposes to the woman j ∈ ℐw at the top of his list. At the end of this stage, some women got proposals (possibly more than one), other women may not get any proposal. 2. At the second stage, each woman who got more than one proposal binds the man whose proposal is most preferable according to her list (who is now engaged). She releases all the other men who proposed. At the end of this stage, the men’s set ℐm is composed of two parts: engaged and released. 3. At the next stage each released man makes a proposal to the next woman in his preference list (whenever she is engaged or not). 4. Back to stage 2. It is easy to verify that this process must end at a finite number of steps. At the end of this process all women and men are engaged. This is a stable matching! Of course, we could reverse the role of men and women in this algorithm. In both cases we get a stable matching. The algorithm we indicated is the one which is best from the men’s point of view. Of course, the reversed case is best for the women, in fact (e. g., [35, 22]). Theorem 2.1. For any stable matching τ the rank of the woman τ(i) according to man i is at most the rank of the woman matched to i by the above men proposing algorithm. Example 2.1.1. men preference – w1 ≻1 w2 ≻1 w3 – w3 ≻1 w2 ≻1 w1 – w1 ≻1 w2 ≻1 w3 women preference – m2 ≻1 m3 ≻1 m1 – m1 ≻1 m2 ≻1 m3 – m1 ≻1 m2 ≻1 m3 m1 m m m m m ) ( 2) ( 3) ⇒ ( 1) ( 2) ( 3) w1 w3 w1 w2 w3 w1 m2 m1 m1 m2 m1 m m m m Women propose: ( ) ( ) ( ) ⇒ ( ) ( ) ( 2 ) ⇒ ( 2 ) ( 1 ) ( 3 ) w1 w2 w3 w1 w2 w3 w1 w2 w3

Men propose: (

2.2 Where money comes in...

| 19

In particular, we obtain the following. Theorem 2.2. A stable matching always exists.

2.2 Where money comes in... Assume that we can guarantee a “cut” ui for each married man i and a cut vj for each married woman j (both in, say, US dollars). In order to define a stable marriage we have to impose some conditions which will guarantee that no man or woman can increase his or her cut by marrying a different partner. For this let us define, for each pair (i, j), a bargaining set F(i, j) ⊂ ℝ2 which contains all possible cuts (ui , vj ) for a matching of man i with woman j. Assumption 2.2.1. i) For each i ∈ ℐm and j ∈ ℐw , F(i, j) are closed sets in ℝ2 . Let F0 (i, j) be the interior of F(i, j). ii) F(i, j) is monotone in the following sense: If (u, v) ∈ F(i, j), then (u󸀠 , v󸀠 ) ∈ F(i, j) whenever u󸀠 ≤ u and v󸀠 ≤ v. iii) There exist C1 , C2 ∈ ℝ such that {(u, v); max(u, v) ≤ C2 } ⊂ F(i, j) ⊂ {(u, v); u + v ≤ C1 } for any i ∈ ℐm , j ∈ ℐw . The meaning of the feasibility set is as follows. Any married couple (i, j) ∈ ℐm × ℐw can guarantee the cut u for i and v for j, provided (u, v) ∈ F (i, j).

Definition 2.2.1. A matching τ : ℐm → ℐw is stable iff there exists a vector (u1 , . . . , uN , v1 , . . . , vN ) ∈ ℝ2N such that (ui , vj ) ∈ ℝ2 − F0 (i, j) for any (i, j) ∈ ℐm × ℐw and (ui , vτ(i) ) ∈ F(i, τ(i)) for any i ∈ ℐw . We now demonstrate that Definition 2.2.1 is a generalization of stable marriage in the non-transferable case, as described in Section 2.1 above. For this we quantify the preference list introduced in (2.1), (2.2). Assume that a man i ∈ ℐm will gain the cut θm (i, j) if he marries woman j ∈ ℐw . So, the vector θm (i, 1), . . . , θm (i, N) is a numeration of (2.1). In particular, j1 ≻i j2 iff θm (i, j1 ) > θm (i, j2 ). Likewise, we associate a cut θw (i, j) for a woman j ∈ ℐw marrying a man i ∈ ℐm , such that i1 ≻j i2 iff θm (i, j1 ) > θm (i, j2 ). Define the feasibility sets F(i, j) := {u ≤ θm (i, j); v ≤ θw (i, j)};

(2.3)

20 | 2 The stable marriage problem

Figure 2.1: Pairwise bargaining sets.

see Figure 2.1(a). Suppose now τ is a stable matching according to Definition 2.2.1. Let (u1 , . . . , uN , v1 , . . . , vN ) be as given in Definition 2.2.1. We obtain that for any man i, (ui , vτ(i) ) ∈ F(i, τ(i)), which by (2.3) is equivalent to ui ≤ θm (i, τ(i)) and vτ(i) ≤ θw (i, τ(i)). Likewise, for any woman j, (uτ−1 (j) , vj ) ∈ F(τ−1 (j), j), which by (2.3) is equivalent to uτ−1 (j) ≤ θm (τ−1 (j), j) and vj ≤ θw (τ−1 (j), j). If j ≠ τ(i), then by definition again, (ui , vj ) ∈ ̸ F0 (i, j), which means by (2.3) that either ui ≥ θm (i, j) and/or vj ≥ θw (i, j). Hence either θm (i, τ(i)) ≥ θm (i, j) and/or θw (τ−1 (j), j) ≥ θw (i, j). According to our interpretation it means that either man i prefers woman τ(i) over j, or woman j prefers man τ−1 (j) over i. That is, (i, j) is not a blocking pair.

2.3 Marriage under sharing In the case we allow sharing (transferable utility) we assume that each married couple may share their individual cuts. Thus, if θm (i, j), θw (i, j) are as defined in the nontransferable case above, man i can transfer a sum w to woman j (in order to prevent a gender bias we assume that w can be negative as well). Thus, the man’s cut from this marriage is θm (i, j)−w, while the woman’s cut is θw (i, j)+w. Since we do not prescribe w, the feasibility set for a pair (i, j) takes the form F(i, j) := {(u, v) : u + v ≤ θ(i, j)},

(2.4)

2.4 General case

| 21

where θ(i, j) := θm (i, j) + θw (i, j) (Figure 2.1(b)). The definition of a stable marriage in the transferable case is implied from Definition 2.2.1 in this special case. Definition 2.3.1. A matching τ is stable iff there exists (u1 , . . . , uN , v1 , . . . , vN ) ∈ ℝ2N such that ui + vτ(i) = θ(i, τ(i)) for any i, and ui + vj ≥ θ(i, j) for any i, j. It turns out that there are several equivalent definitions of stable marriages in the sense of Definition 2.3.1. Here we introduces three of these. Theorem 2.3. τ is a stable marriage in the sense of Definition 2.3.1 iff one of the following equivalent conditions is satisfied: i) Optimality: There exists (u01 , . . . , vN0 ) ∈ ℝ2N satisfying 0 u0i + vτ(i) = θ(i, τ(i))

for any i ∈ ℐm which minimizes ∑i∈ℐ (ui + vi ) over the set W := {(u1 , . . . , vN ) ∈ ℝ2N ; ui + vj ≥ θ(i, j) ∀(i, j) ∈ ℐm × ℐw }. ii) Efficiency: τ maximizes ∑Ni=1 θ(i, σ(i)) on the set of all matchings σ : ℐm → ℐw . iii) Cyclic monotonicity: For any chain i1 , . . . , ik ∈ {1, . . . , N}, the inequality k

∑ (θ(ij , τ(ij )) − θ(ij , τ(ij+1 )) ≥ 0 j=1

(2.5)

holds, where ik+1 = i1 . In particular, we have the following corollary. Corollary 2.3.1. A stable matching according to Definition 2.3.1 always exists. For the proof of Theorem 2.3, see Section 2.6. In fact, the reader may, at this point, skip Sections 2.4–2.5 and Chapter 3 as the rest of the book is independent of these.

2.4 General case In the general case of Assumption 2.2.1, the existence of a stable matching follows from the following theorem. Theorem 2.4. Let V ⊂ ℝ2N be defined as follows: (u1 , . . . , uN , v1 , . . . , vN ) ∈ V ⇔ ∃ an injection τ : ℐm → ℐw such that (ui , vτ(i) ) ∈ F(i, τ(i)) ∀ i ∈ ℐm .

22 | 2 The stable marriage problem Then there exists (u1 , . . . , uN , v1 , . . . , vN ) ∈ V such that (ui , vj ) ∈ ℝ2 − F0 (i, j)

(2.6)

for any (i, j) ∈ ℐm × ℐw . The set of vectors in V satisfying (2.6) is called the core. Definition 2.2.1 can now be recognized as the non-emptiness of the core, which is equivalent to the existence of a stable matching. Theorem 2.4 is, in fact, a special case the celebrated theorem of Scarf [42] for cooperative games, tailored to the marriage scenario. As we saw, it can be applied to the fully non-transferable case (2.3), as well as to the fully transferable case (2.4). There are other, sensible models of partial transfers which fit into the formalism of Definition 2.2.1 and Theorem 2.4. Let us consider several examples: 1. Transferable marriages restricted to non-negative cuts: In the transferable case the feasibility sets may contain negative cuts for man u or for woman v (even though not for both, if it is assumed that θ(i, j) > 0). To avoid the undesired stable marriages where one of the partners gets a negative cut we may replace the feasibility set (2.4) by F(i, j) := {(u, v) ∈ ℝ2 ; u + v ≤ θ(i, j), u ≤ θ(i, j), v ≤ θ(i, j)}

2.

(Figure 2.1(c)). It can be easily verified that if (u1 , . . . , vN ) ∈ V contains negative components, then ([u1 ]+ , . . . , [vN ]+ ), obtained by replacing the negative components by 0, is in V as well. Thus, the core of this game contains vectors in V of non-negative elements. In the transferable case (2.4) we allowed both men and women to transfer money to their partner. Indeed, we assumed that the man’s i cut is θm (i, j) − w and the woman’s j cut is θw (i, j) + w, where w ∈ ℝ. Suppose we wish to allow only transfer between men to women, so we insist on w ≥ 0.1 In that case we choose (Figure 2.1(d)) F(i, j) := {(u, v) ∈ ℝ2 ; u + v ≤ θ(i, j); u ≤ θm (i, j)}.

3.

Let us assume that the transfer w from man i to woman j is taxed, and the tax depends on i, j. Thus, if man i transfers w > 0 to a woman j he reduces his cut by w, but the woman cut is increased by an amount βi,j w, where βi,j ∈ [0, 1]. Here 1 − βi,j is the tax implied for this transfer. It follows that ui ≤ θm (i, j) − w; vj ≤ θw (i, j) + βi,j w, w ≥ 0.

1 Of course we could make the opposite assumption w ≤ 0. We leave the reader to change this example according to his view on political correctness...

2.5 Stability by fake promises | 23

Hence −1 F(i, j) := {(u, v) ∈ ℝ2 ; ui + βi,j vj ≤ θβ (i, j), ui ≤ θm (i, j)}, −1 where θβ (i, j) := θm (i, j) + βi,j θw (i, j). This is demonstrated by Figure 2.1(d), where the dashed line is tilted.

2.5 Stability by fake promises We now describe a different notion of stability. Suppose a man can make a promise to a married woman (who is not his wife), and vice versa. The principle behind it is that each of them does not intend to honor his/her own promise, but, nevertheless, believes that the other party will honor her/his promise. It is also based on some collaboration between the set of betraying couples. For simplicity of presentation we assume that the matching τ is given by “the identity” τ(i) = i, where i ∈ ℐm represents a man and i = τ(i) ∈ ℐw represents the matched woman. Evidently, we can always assume this by ordering the list of men (or women) in a different way. Let us repeat the definition of stability in the context of non-transferable matching (Definition 2.1.1). For this, we recall the definition of a blocking pair (i, j): θm (i, j) > θm (i, i)

and θw (i, j) > θw (j, j),

which we rewrite as Δ(0) (i, j) := min{θm (i, j) − θm (i, i), θw (i, j) − θw (j, j)} > 0.

(2.7)

Assume that a man i ∈ ℐm can offer some bribe b to any other woman j he might be interested in (except his own wife, so j ≠ i). His cut for marrying j is now θm (i, j)−b. The cut of the woman j should have been θw (i, j) + b. However, the happy woman should pay some tax for accepting this bribe. Let q ∈ [0, 1] be the fraction of the bribe she can get (after paying her tax). Her supposed cut for marrying i is just θw (i, j) + qb. Woman j will believe and accept the offer from man i if two conditions are satisfied: the offer should be both 1. competitive, namely, θw (i, j) + qb ≥ θw (j, j), and 2. trusted, if woman j believes that man i is motivated. This implies θm (i, j) − b ≥ θm (i, i). The two conditions above can be satisfied, and the offer is acceptable, if q(θm (i, j) − θm (i, i)) + θw (i, j) − θw (j, j) > 0.

(2.8)

24 | 2 The stable marriage problem Symmetrically, man i will accept an offer from a woman j ≠ i if q(θw (i, j) − θw (i, i)) + θm (i, j) − θm (j, j) > 0.

(2.9)

Let us define the utility of the exchange i ↔ j: Δ(q) (i, j) := min {

q(θm (i, j) − θm (i, i)) + θw (i, j) − θw (j, j) }, q(θw (i, j) − θw (j, j)) + θm (i, j) − θm (i, i)

(2.10)

so, a blocking-q pair (i, j) is defined by the condition that the utility of exchange is positive for both parties: Δ(q) (i, j) > 0.

(2.11)

Evidently, if q = 0 there is no point of bribing, so a blocking pair corresponding to (2.11) is equivalent to condition (2.7) for the non-transferable case, as expected. For the other extreme case (q = 1) where the bribe is not penalized, the expected profit of both i, j is the same, and equals Δ(1) (i, j) = θm (i, j) − θm (i, i) + θw (i, j) − θw (j, j).

(2.12)

We now consider an additional parameter p ∈ [0, 1] and define the real-valued function on ℝ x 󳨃→ [x]p := [x]+ − p[x]− .

(2.13)

Note that [x]p = x for any p if x ≥ 0, while [x]1 = x for any real x. Definition 2.5.1. Let 0 ≤ p, q ≤ 1. The matching τ(i) = i is (p, q)-stable if for any k ∈ ℕ and i1 , i2 , . . . , ik ∈ {1, . . . , N} k

∑ [Δ(q) (il , il+1 )]p ≤ 0, l=1

where ik+1 = i1 .

What does this mean? Within the chain of pair exchanges (i1 , i1 ) → (i1 , i2 ), . . . , (ik−1 , ik−1 ) → (ik−1 , ik ), (ik , ik ) → (ik , i1 ) each of the pair exchanges (il , il ) → (il , il+1 ) yields a utility Δ(q) (il , il+1 ) for the new pair. The lucky new pairs in this chain of couple exchanges are those who make a positive utility. The unfortunate new pairs are those whose utility is non-positive. The lucky pairs, whose interest is to activate this chain, are ready to compensate the unfortunate ones by contributing some of their gained utility. The chain will be activated (and the original marriages will break down) if the mutual contribution of the fortunate pairs

2.5 Stability by fake promises | 25

is enough to cover at least the p-part of the mutual loss of utility of the unfortunate pairs. This is the condition ∑

Δ(q) (il ,il+1 )>0

Δ(q) (il , il+1 ) + p

k

∑

Δ(q) (il ,il+1 ) 0. l=1

Definition 2.5.1 guarantees that no such chain is activated.

– –

In order to practice this definition, let us look at the extreme cases: p = 0, q = 0. In particular, there is no bribing: A (0, 0)-stable marriage is precisely the stability in the non-transferable case introduced in Section 2.1. p = q = 1. Definition 2.5.1 implies stability if and only if k

∑ Δ(1) (il , il+1 ) ≤ 0

(2.14)

l=1

for any k-chain and any k ∈ ℕ. Let θ(i, j) := θm (i, j) + θw (i, j). Then, (2.10) implies that (2.14) is satisfied if and only if k

∑ θ(il , il+1 ) − θ(il , il ) ≤ 0, l=1

where ik+1 = i1

(2.15)

(check it!). By point (iii) of Theorem 2.3 and Corollary 2.3.1 we obtain the (not really surprising) result. Corollary 2.5.1. A matching is (1,1)-stable iff it is stable in the completely transferable case (2.4). In particular, there always exists a (1, 1)-stable matching. The observation (2.7) and the definition [x]0 := [x]+ imply, together with Theorem 2.2, the following. Corollary 2.5.2. A matching is (0,0)-stable iff it is stable in the non-transferable case (2.3). In particular, there always exists a (0, 0)-stable matching. We now point out the following observation. Theorem 2.5. If τ is (p, q)-stable, then τ is also (p󸀠 , q󸀠 )-stable for p󸀠 ≥ p and q󸀠 ≤ q. ing.

The proof of this theorem follows from the definitions (2.10), (2.13) and the follow-

Lemma 2.1. For any i ≠ j and 1 ≥ q > q󸀠 ≥ 0, 󸀠

(1 + q)−1 Δ(q) (i, j) > (1 + q󸀠 )−1 Δ(q ) (i, j).

26 | 2 The stable marriage problem Proof. For a, b ∈ ℝ and r ∈ [0, 1] define r 1 Δr (a, b) := (a + b) − |a − b|. 2 2 Observe that Δ1 (a, b) ≡ min(a, b). In addition, r 󳨃→ Δr (a, b) is monotone not increasing in r. A straightforward calculation yields min(qa + b, qb + a) = Δ1 (qa + b, qb + a) = (q + 1)Δ 1−q (a, b), 1+q

and the lemma follows from the above observation, upon inserting a = θm (i, j)−θm (i, i) and b = θw (i, j) − θw (j, j). What can be said about the existence of s (p, q)-stable matching in the general case? Unfortunately, we can prove now only a negative result. Proposition 2.1. For any 1 ≥ q > p ≥ 0, a stable marriage does not exist unconditionally. Proof. We only need to present a counter-example. So, let N = 2. To show that the matching τ(1) = 1, τ(2) = 2 is not stable we have to show [Δ(q) (1, 2)]p + [Δ(q) (2, 1)]p > 0,

(2.16)

while to show that τ(1) = 2, τ(2) = 1 is not stable we have to show [Δ(q) (1, 1)]p + [Δ(q) (2, 2)]p > 0.

(2.17)

By the definition (2.10) and Lemma 2.1, Δ(q) (1, 2) = (q + 1)Δr (θm (1, 2) − θm (1, 1), θw (1, 2) − θw (2, 2)) , Δ(q) (2, 1) = (q + 1)Δr (θm (2, 1) − θm (2, 2), θw (2, 1) − θw (1, 1)) , where r = so

1−q . To obtain Δ(q) (1, 1), Δ(q) (2, 2) we just have to exchange man 1 with man 2, 1+q

Δ(q) (2, 2) = (q + 1)Δr (θm (2, 2) − θm (2, 1), θw (2, 2) − θw (1, 2)) , Δ(q) (1, 1) = (q + 1)Δr (θm (1, 1) − θm (1, 2), θw (1, 1) − θw (2, 1)) . All in all, we only have four parameters to play with: a1 := θm (1, 2) − θm (1, 1), b1 = θm (2, 1) − θm (2, 2),

a2 = θw (1, 2) − θw (2, 2),

b2 = θw (2, 1) − θw (1, 1),

so the two conditions to be verified are [Δr (a1 , a2 )]p + [Δr (b1 , b2 )]p > 0;

[Δr (−a1 , −b2 )]p + [Δr (−b1 , −a2 )]p > 0.

2.6 The discrete Monge problem

| 27

Let us insert a1 = a2 := a > 0; b1 = b2 := −b, where b > 0. So [Δr (a1 , a1 )]p = a,

[Δr (b1 , b2 )]p = −pb,

while Δr (−a1 , −b2 ) = Δr (−b1 , −a2 ) = b−a − 2r (a + b). In particular, the condition ba < 1−r 2 1+r implies [Δr (−a1 , −b2 )]p = [Δr (−b1 , −a2 )]p > 0, which verifies (2.17). On the other hand, 1−r > p. Recalling if a − pb > 0, then (2.16) is verified. Both conditions can be verified if 1+r 1−r q = 1+r we obtain the result. Based on Theorem 2.5 and Proposition 2.1 we propose the following. Conjecture. There always exists a stable (p, q)-marriage iff q ≤ p (Fig. 2.2).

Figure 2.2: Conjecture: Is there an unconditional existence of stable marriages in the gray area.

2.6 The discrete Monge problem In Theorem 2.3 we now encounter, for the first time, the Monge problem in its discrete setting. Let {θ(i, j)} be an N × N matrix of rewards. The reward of a given bijection τ : ℐm ↔ ℐw is defined as N

θ(τ) := ∑ θ(i, τ(i)). i=1

(2.18)

Definition 2.6.1. A bijection τ is a Monge solution with respect to {θ} if it maximizes τ 󳨃→ θ(τ) among all bijections. Theorem 2.3 claims, in particular, that τ is a Monge solution iff it is a stable marriage with respect to transferable utility (2.4). To show it we first establish the equivalence between Monge solutions (ii) to cyclically monotone matching, as defined in part (iii) of this theorem.

28 | 2 The stable marriage problem Again we may assume, with no limitation of generality, that τ(i) = i is a Monge solution, namely, N

N

i=1

i=1

∑ θ(i, i) ≥ ∑ θ(i, σ(i)) for any other matching σ. Given a k-chain {i1 , . . . , ik }, consider the associated cyclic permutation σ(i1 ) = i2 , . . . , σ(ik−1 ) = ik , σ(ik ) = i1 . Then θ(σ ∘ τ) ≤ θ(τ) by definition. On the other hand, θ(τ) − θ(σ ∘ τ) is precisely the left side of (2.5) k

∑ θ(ij , ij ) − θ(ij , ij+1 ) ≥ 0. j=1

In the opposite direction: let − u0i :=

inf

k−1

k−chains,k∈ℕ

( ∑ θ(il , il ) − θ(il+1 , il )) + θ(ik , ik ) − θ(i, ik ). l=1

(2.19)

Let α > −u0i and consider a k-chain realizing k−1

α > ( ∑ θ(il , il ) − θ(il+1 , il )) + θ(ik , ik ) − θ(i, ik ). l=1

(2.20)

By cyclic monotonicity, ∑kl=1 θ(il , il ) − θ(il+1 , il ) ≥ 0. Since ik+1 = i1 , k−1

∑ θ(il , il ) − θ(il+1 , il ) ≥ θ(i1 , ik ) − θ(ik , ik ), l=1

so (2.20) implies α > θ(i1 , ik ) − θ(i, ik ) ≥ 0, in particular, u0i < ∞. Hence, for any j ∈ ℐm , k−1

α + θ(i, i) − θ(j, i) > ( ∑ θ(il , il ) − θ(il+1 , il )) + θ(ik , ik ) − θ(i, ik ) + θ(i, i) − θ(j, i) ≥ −u0j , l=1

(2.21)

where the last inequality follows by the substitution of the (k +1)-cycle i, i1 . . . ik (where ik+1 = i) in (2.19). Since α is any number bigger than −u0i it follows that − u0i + θ(i, i) − θ(j, i) ≥ −u0j .

(2.22)

2.6 The discrete Monge problem

| 29

To prove that the Monge solution is stable, we define vj0 := θ(j, j) − u0j , so u0j + vj0 = θ(j, j).

(2.23)

Then (2.22) implies (after interchanging i and j) u0i + vj0 = u0i + θ(j, j) − u0j ≥ u0i − u0i + θ(i, j) = θ(i, j)

(2.24)

for any i, j. Thus, (2.23), (2.24) establish that τ(i) = i is a stable marriage via Definition 2.3.1. Finally, to establish the equivalence of the optimality condition (i) in Theorem 2.3 to condition (ii) (Monge solution), we note that for any (u1 , . . . , vN ) ∈ W, ∑Ni=1 ui + vi ≥ ∑Ni=1 θ(i, i), while (u01 , . . . , vN0 ) calculated above is in W and satisfies the equality.

3 Many to few: stable partitions The employer generally gets the employees he deserves. (J. Paul Getty)

3.1 A non-transferable partition problem We now abandon the gender approach of Chapter 2. Instead of the men–women groups ℐm , ℐw , let us consider a set ℐ of N agents (firms) and a set of consumers (employees) X. We do not assume, as in Chapter 2, that the two sets are of equal cardinality. In fact, we take the cardinality of X to be much larger than that of ℐ . It can also be (and in general is) an infinite set. Let us start from the ordinal viewpoint: We equip X with a σ-algebra ℬ ⊂ 2X such that X ∈ ℬ as well as, for any x ∈ X, {x} ∈ ℬ, and an atomless, positive measure μ: (X, ℬ, μ); μ : ℬ → ℝ+ ∪ {0}.

(3.1)

In addition, we consider the structure of preference list generalizing (2.1), (2.2): Each firm i ∈ ℐ orders the potential employees X according to a strict preference list. Let ≻i be a strict, measurable-order relation on X. See the following definition. Definition 3.1.1. i) Non-symmetric: for any x ≠ y either x ≻i y or y ≻i x (but not both). ii) Transitive: x ≻i y, y ≻i z implies x ≻i z for any distinct triple x, y, z ∈ X. iii) For all y ∈ X, i ∈ ℐ , Ai (y) := {x ∈ X; x ≻i y} ∈ ℬ. iv) If x1 ≻i x2 , then μ (y; (y ≻i x2 ) ∩ (x1 ≻i y)) > 0. In addition, for any x ∈ X we also assume the existence of order relation ≻x on ℐ such that the following holds. Definition 3.1.2. i) Non-symmetric: for any i ≠ j either i ≻x j or j ≻x i (but not both). ii) Transitive: i ≻x j, j ≻x k implies i ≻x k for any distinct triple i, j, k ∈ ℐ . iii) For all i ≠ j ∈ ℐ , {x ∈ X; i ≻x j} ∈ B. Thus: The firm i prefers to hire x ∈ X over y ∈ X iff x ≻i y. Likewise, a candidate x ∈ X prefers firm i over j as a employer iff i ≻x j.

What is the extension of a bijection τ : ℐm ↔ ℐw to that case? Since the cardinality of X is larger than that of ℐ , there are no such bijections. We replace the bijection τ by a measurable mapping τ : X → ℐ . https://doi.org/10.1515/9783110635485-003

32 | 3 Many to few: stable partitions We can think about such a surjection as a partition A⃗ := {Ai ∈ ℬ, i ∈ ℐ , Ai ∩ Aj = 0 if i ≠ j, ⋃ Ai = X}, i∈ℐ

where Ai := τ−1 (i). We also consider cases where τ is not a surjection, so there are unemployed people A0 and ⋃i∈ℐ Ai ⊂ X. Another assumption we make is that the capacity of the firms can be limited. That is, for any firm i ∈ ℐ , the number of its employees is not larger than some mi > 0: μ(Ai ) ≤ mi . Note that we do not impose any condition on the capacities mi (except positivity). In particular, ∑i∈ℐ mi can be smaller than, equal to, or bigger than μ(X). Evidently, if ∑i∈ℐ mi < μ(X), then there is an unemployed set of positive measure. Let us define a “fictitious firm” {0} which contains all unfortunate candidates which are not accepted by any firm. The order relation (X, ≻x ) is extended to ℐ ∪ {0} as i ≻x 0 for any i ∈ ℐ and any x ∈ X (i. e., we assume that anybody prefers an employment by any firm over unemployment). Definition 3.1.3. Let m⃗ := (m1 , . . . , mN ) ∈ ℝN+ . Let A⃗ := (A1 , . . . , AN ) be a subpartition and A0 := X − ⋃i∈ℐ Ai . ⃗ Such a subpartition A⃗ is called an m-subpartition if μ(Ai ) ≤ mi for any i ∈ ℐ , and μ(A0 ) = 0 ∨ {μ(X) − ∑i∈ℐ mi }. Definition 3.1.4. A subpartition is called stable if, for any i ≠ j, i, j ∈ ℐ ∪ {0} and any x ∈ Ai , either i ≻x j or y ≻j x for any y ∈ Aj . ⃗ Theorem 3.1. For any m⃗ ∈ ℝN+ there exists a stable m-subpartition. The proof of this theorem, outlined in Section 3.1.1 below, is a constructive one. It is based on a generalization of the Gale–Shapley algorithm, described in Section 2.1.1. For describing this algorithm we need few more definitions: For any i ∈ ℐ and A ∈ ℬ, the set Ci(1) (A) ∈ ℬ is the set of all candidates in A whose i is the first choice: Ci(1) (A) := {x ∈ A; ∀j ≠ i, i ≻x j}. By recursion we define Ci(k) (A) to be the set of employees in A such that i is their k-choice: Ci(k) (A) := {x ∈ A; ∃ℐk−1 ⊂ ℐ ; i ∈ ̸ ℐk−1 ; |ℐk−1 | = k − 1; ∀j ∈ ℐk−1 , j ≻x i; ∀j ∈ ℐ − (ℐk−1 ∪ {i}), i ≻x j}. By definition, Ci(k) (A) ∈ ℬ for any i ∈ ℐ , k = 1, . . . , N and A ∈ ℬ.

3.1 A non-transferable partition problem

| 33

3.1.1 The Gale–Shapley algorithm for partitions At the beginning of each step k there is a subset Xk−1 ⊂ X of free candidates. At the beginning of the first step all candidates are free, so X0 := X. At the first stage, each x ∈ X0 applies to the firm at the top of his list. So, at the end of this stage, each firm i gets an employment request from Ci(1) (X0 ) (which, incidentally, can be empty). At the second part of the first stage, each firm evaluates the number of requests she got. If μ(Ci(1) (X0 )) < mi she keeps all candidates and we define A(1) := Ci(1) (X0 ). i Otherwise, she ejects all less favorable candidates until she fills her quota mi : Let (1) (1) A(1) i := ⋃ {Ai (y) ∩ Ci (X0 ); μ(Ai (y) ∩ Ci (X0 )) ≤ mi } , y∈X

where Ai (y) is as in Definition 3.1.1(iii). Note that A(1) ∈ ℬ. Indeed, let α(y) := μ(Ai (y) ∩ Ci(1) (X0 )) and i mi := sup{α(y); α(y) ≤ mi }. y∈X

Then there exists a sequence yn ∈ X such that α(yn ) is monotone non-decreasing and lim α(yn ) = mi . We obtain (1) A(1) i ≡ ⋃ {Ai (yn ) ∩ Ci (X0 )} , n

are both in ℬ. ∈ ℬ since Ai (yn ) and The set of candidates who where rejected at the end of the first step is the set of free candidates so

A(1) i

Ci(1) (X0 )

X1 := X − ⋃ A(1) i . i∈ℐ

At the (k + 1)th stage we consider the set of free candidates Xk as the set who were rejected at the end of the k stage. Each employee in Xk was rejected n times, for some 1 ≤ n ≤ k. So each x ∈ Xk who was rejected n times proposes to the firm i if i is the next (n + 1)th firm on their priority list, that is, if x ∈ Ci(n+1) (X). Note that for any such person there exists a chain 1 ≤ l1 < l2 < ⋅ ⋅ ⋅ < ln = k such that x ∈ ⋂Xlj − ( j≤n

⋃

1≤q ϕj (x), and x ≻i y by ψi (x) > ψi (y). However, the cases ϕi (x) = ϕj (y) and ψi (x) = ψi (y) violate condition (i) in Definitions 3.1.1 and 3.1.2. For ≻x , ≻i to be consistent with these definitions we omit from the set X all points for which there is an equality of ψi (x) = ψj (x) or ϕi (x) = ϕi (y). Let Δ1 (X) := {x ∈ X; ∃y ≠ x, i ∈ ℐ , ϕi (x) = ϕi (y)}, Δ2 (X) := {x ∈ X; ∃i ≠ j ∈ ℐ , ψi (x) = ψj (x)},

and define X0 := X − (Δ1 (X) ∪ Δ2 (X)). Then ∀x, y ∈ X0 , i, j ∈ ℐ ; i ≻x j iff ϕi (x) > ϕj (x), x ≻i y iff ψi (x) = ψj (y).

(3.2)

As in Section 3.1 we consider the “null firm” {0} and ϕ0 (x) = 0 for all x ∈ X, while ϕi (x) > 0 for any x ∈ X, i ∈ ℐ . Under the above definition, the non-transferable partition model is obtained under the following definition of feasibility sets: F(i, x) := {(u, v); u ≤ ψi (x) v ≤ ϕi (x)} ,

i ∈ ℐ , x ∈ X0 ,

(3.3)

where ϕi , ψi are assumed to be strictly positive, measurable functions on X0 . The exis⃗ tence of a stable m-partition under (3.3) is, then, guaranteed by Theorem 3.1. The case where firms and employees share their utilities is a generalization of (2.4):

36 | 3 Many to few: stable partitions

F (i, x) := {(u, v); u + v ≤ θi (x)} , where

(3.4)

θi (x) := ϕi (x) + ψi (x).

⃗ The existence of stable m-partitions in the transferable case (3.4), and its generalization, is the main topic of this book! We may also attempt to generalize the notion of q-blocking pairs with respect to a partition A⃗ ∈ 𝒫 N . In analogy to (2.10), (x, y) is a blocking pair if x ∈ Ai , y ∈ Aj and Δ(q) (x, y) > 0, where Δ(q) (x, y) := min {

q(ψj (x) − ψi (x)) + ϕj (x) − ϕj (y) }. q(ϕj (x) − ϕj (y)) + ψj (x) − ψi (x)

(3.5)

Definition 2.5.1 is generalized as follows. Definition 3.2.2. Given a partition A,⃗ a k-chain is a sequence xi1 . . . xik , where xil ∈ Ail for any 1 ≤ l ≤ k, and xik = xi1 (in particular, ik = i1 ). A partition A⃗ in X0 is (p, q)-stable if for any k ∈ ℕ, any k-chain k

∑ [Δ(q) (xil , xil+1 )]p ≤ 0, l=1

where [⋅]p is as defined in (2.13). What does it mean? Again let us assume first q = 0 (no bribing) and p = 0 (no sharing). Then we have the following. A partition A⃗ is (0, 0)-unstable iff there exist x ∈ Ai , y ∈ Aj , i ≠ j for which Δ(0) (x, y) > 0. This implies that ψj (x) > ψi (x) and, in addition, ϕj (x) > ϕj (y). Surely x will prefer agent j over his assigned agent i, and agent j will prefer x over his assigned customer y as well. So, j will kick y out and x will join j instead, for the benefit of both x and j.

In particular, we have the following. Any stable (0, 0)-subpartition is a stable subpartition in the sense of Definition 3.1.4.

What about the other extreme case p = q = 1? It implies (using ik+1 = i1 ) k

k

l=1

l=1

∑ Δ(1) τ (xil , xil+1 ) ≡ ∑ ψil (xil+1 ) − ψil+1 (xil+1 ) + ϕil (xil+1 ) − ϕil (xil ) k

≡ ∑ θil (xil+1 ) − θil+1 (xil+1 ) ≤ 0, l=1

(3.6)

3.2 Transferable utilities | 37

where θi is as defined in (3.4). Let us define A⃗ to be θ-cyclic monotone iff for any k ∈ ℕ and k-chain (i1 , . . . , ik ) in ℐ and any xi1 ∈ Ail , 1 ≤ l ≤ k, k

∑ θil (xil+1 ) − θil (xil ) ≤ 0. l=1

In the complete cooperative economy, where a firm i and an employee x share their utilities θi (x) = ψi (x) + ϕi (x), a partition A⃗ is (1, 1)-stable iff it is cyclical monotone. This means that: Not all members of any given chain of replacements xi1 → xi2 , xi2 → xi3 . . . xik → xik+1 ≡ xi1 , xil ∈ Ail , will gain utility, even if the other members are ready to share their benefits (and losses) among themselves.

The connection between a stable partition in the (1, 1)-sense and the (3.4)-sense is not evident. In the next chapter we discuss this subject in some detail.

4 Monge partitions The purpose of a business is to create a customer. (Peter Drucker)

We pose some structure on X and the utility functions θi . Standing Assumption 4.0.1. i) X is a compact topological space. ii) The N utility functions θ1 , . . . , θN : X → ℝ are continuous. We find it convenient to change the interpretation of candidates/firms of Chapter 3 as follows: The set X is the set of customers (or consumers), and the set ℐ is the set of agents (or experts). The function θi : X → ℝ represents the “utility” of agent i, namely, θi (x) is the surplus of the coupling of x to i. Definition 4.0.1. An open N subpartition of X is a collection of N disjoint open subsets of X. We denote the collection of all such subpartitions by 𝒪𝒮𝒫

N

:= {A⃗ = (A1 , . . . , AN ), Ai is an open subset of X, Ai ∩ Aj = 0 if i ≠ j} .

For any A⃗ ∈ 𝒪𝒮𝒫 N we denote A0 := X − ⋃i∈ℐ Ai . Definition 4.0.2. An open subpartition A⃗ is stable iff it is cyclically monotone with respect to A0 , A1 , . . . , AN , i. e., for any k ∈ ℕ and any k-chain xi1 , . . . , xik , il ∈ ℐ ∪ {0}, where xil is an interior point of Ail , 1 ≤ l ≤ k, k

∑ θil (xil ) − θil+1 (xil ) ≥ 0. l=1

(4.1)

Here θik+1 := θii and θ0 ≡ 0. Note that since Ai are open sets for i ∈ ℐ , the condition “xil is an interior point of Ail ” simply means xil ∈ Ail if il ≠ 0. If A0 has a null interior, then we only consider chains in ℐ .

4.1 Capacities Here we assume that the agents have a limited capacity. This symbolizes the total number of consumers each agent can serve. For this we define an additional structure on the set X. Standing Assumption 4.1.1. We assume ℬ is the Borel σ-algebra corresponding to the assumed topology of X; μ ∈ ℳ+ (X) is a given positive, regular, and atomless Borel measure on (X, ℬ), and X = supp(μ). https://doi.org/10.1515/9783110635485-004

40 | 4 Monge partitions Let us recall that if μ is a regular positive Borel measure on X, then for any A ∈ ℬ, A ≠ 0 and any ϵ > 0 there exist an open U ⊃ A and a compact K ⊂ A such that μ(U) ≤ μ(A) + ϵ,

μ(K) ≥ μ(A) − ϵ.

An atom of μ is a point x ∈ X for which μ({x}) > 0. An atomless measure contains no atoms. Recall that supp(μ) is a closed set, obtained as the intersection of all compact sets K ⊆ X for which μ(K) = μ(X). The measure μ represents the distribution of the consumers: for A ∈ ℬ, μ(A) stands for the number of consumers in A (not necessarily an integer). The meaning of a limited capacity mi > 0 for an agent i is μ(Ai ) ≤ mi . The set of open subpartitions subjected to a given capacity m⃗ := (m1 , . . . , mN ), mi ≥ 0 is denoted by N

N

𝒪𝒮𝒫 {≤m}⃗ = {A⃗ := (A1 , . . . , AN ) ∈ 𝒪𝒮𝒫 ; μ(Ai ) ≤ mi } .

(4.2)

More generally: For any closed set K ⊂ ℝN+ , N

N

𝒪𝒮𝒫 K := {A⃗ := (A1 , . . . , AN ) ∈ 𝒪𝒮𝒫 ; (μ(A1 ), . . . , μ(AN )) ∈ K}.

(4.3)

In particular, if ⃗ K = Km⃗ := {m⃗ ≥ s⃗ ≥ 0},

(4.4)

then 𝒪𝒮𝒫 NKm⃗ is reduced to 𝒪𝒮𝒫 N{≤m}⃗ . We distinguish three cases: m⃗ is oversaturated (OS) if ∑ mi > μ(X), i∈ℐ

(4.5)

which means that the supply of the experts surpasses the demand of the consumers; saturated (S) if ∑ mi = μ(X), i∈ℐ

(4.6)

which means that the supply of the experts and the demand of consumers are balanced, and undersaturated (US) if ∑ mi < μ(X), i∈ℐ

(4.7)

which means that the demand of the consumers surpasses the supply of the experts. ⃗ by 𝒪𝒮𝒫 N{m}⃗ , i. e., If m⃗ is either S or US we denote 𝒪𝒮𝒫 NK where K := {m} N

N

𝒪𝒮𝒫 {m}⃗ := {A⃗ := (A1 , . . . , AN ) ∈ 𝒪𝒮𝒫 ; μ(Ai ) = mi }.

(4.8)

4.2 First paradigm: the big brother |

41

4.2 First paradigm: the big brother The “big brother” splits the consumers X between the experts in order to maximize the total surplus, taking into account the capacity constraints. If m⃗ is either US or S, then ⃗ := Σθ (m)

sup

N ⃗ A∈𝒪𝒮𝒫 ⃗ {m}

⃗ θ(A),

(4.9)

where θ(A)⃗ := ∑ ∫ θi (x)dμ i∈ℐ A

(4.10)

i

⃗ = is the total profit conditioned on the partition {Ai }. Note that by this definition, Σθ (m) −∞ if m⃗ is OS.1 More generally, for any closed K ⊂ ℝN+ , Σθ (K) :=

sup

N ⃗ A∈𝒪𝒮𝒫 K

⃗ θ(A)⃗ = sup Σθ (m). ⃗ m∈K

(4.11)

Remark 4.2.1. Let K = Km⃗ (equation 4.4). If m⃗ is S or US and, in addition, the utilities θi are all non-negative on X, then the maximizer A⃗ of (4.11) is also a maximizer of (4.9), i. e., it satisfies μ(Ai ) = mi for any i ∈ ℐ (so A⃗ ∈ 𝒪𝒮𝒫 N{m}⃗ ). What is the relation between maximizers of (4.9) and stable subpartitions (in the sense of Definition 4.0.2)? Proposition 4.1. If A⃗ ∈ 𝒪𝒮𝒫 N{m}⃗ is a maximizer in (4.9), then it is a stable open subpartition. Proof. Let A⃗ be a maximizer of (4.9). If A⃗ is not stable, then by Definition 4.0.2 there exists a chain (xil , Ail ) such that k

∑ θil (xil ) − θil+1 (xil ) < 0. l=1

Since xi are interior points of Ai by assumption and μ is regular, there exist ϵ > 0 and open neighborhoods xi ∋ Ui ⊂ Ai such that μ(Ui ) = μ(Ū i ) = ϵ for any i ∈ ℐ (here Ū is the closure of U). Since θ⃗ are continuous functions we can choose ϵ sufficiently small such that, for some δ > 0, k

∑ θil (x̃il ) − θil+1 (x̃il ) < −δ l=1

1 The supremum over a null set is always −∞.

42 | 4 Monge partitions for any sequence x̃i ∈ Ū i , i ∈ ℐ (again we set x̃ik+1 = x̃i1 ). In particular, k

∑ ∫ [θil − θil+1 ] dμ < −ϵδ. l=1 U

(4.12)

il

Define Bil := (Uil−1 ∪ Ail ) − Ū il for l = 1, . . . , k (recall i0 = ik ), and Bj = Aj if j ∈ ̸ {i1 , . . . , ik }. By definition μ(Bi ) = mi for any i ∈ ℐ , B⃗ := (B1 , . . . , BN ) ∈ 𝒪𝒮𝒫 N{m}⃗ . By (4.12) we obtain θ(B)⃗ := ∑ ∫ θi dμ ≤ θ(A)⃗ − ϵδ, i∈ℐ B

i

contradicting the maximality of θ(A)⃗ on 𝒪𝒮𝒫 Nm⃗ .

4.3 Second paradigm: free market Suppose there is no big brother. The market is free, and each consumer may choose his favorite expert to maximize his own utility. Each expert determines the price she collects for consulting a consumer. Let pi ∈ ℝ be the price requested by expert i, p⃗ := (p1 , . . . , pN ) ∈ ℝN . Remark 4.3.1. A price pi can be positive, negative, or zero. In the second case −pi is a “bonus.” The utility of a consumer x choosing expert i is, therefore, θi (x)−pi , if it is positive. If θi (x) − pi ≤ 0, then the consumer will avoid expert i, so he pays nothing and gets nothing from expert i. The net income of consumer x choosing expert i is, therefore, [θi (x) − pi ]+ := (θi (x) − pi ) ∨ 0. Since any consumer wishes to maximize his income, we obtain the income of any consumer x ∈ X by ξ + (p,⃗ x) := max[θi (x) − pi ]+ . i∈ℐ

(4.13)

The set of consumers who give up counseling by any of the experts is A+0 (p)⃗ = {x ∈ X; θi (x) − pi < 0 for any i ∈ ℐ },

(4.14)

while the set of consumers who prefer expert i is, then, ⃗ A+i (p)⃗ := {x ∈ X; θi (x) − pi ≥ θj (x) − pj ∀j ∈ ℐ } − A+0 (p). Let ⃗ . . . , A+N (p)) ⃗ . A(⃗ p)⃗ := (A+1 (p),

(4.15)

4.4 The big brother meets the free market | 43

Note that the sets A+i (p)⃗ are not necessarily disjoint (for i ∈ ℐ ) nor open. So A(⃗ p)⃗ ∈ ̸ 𝒪𝒮𝒫 N , in general. We denote A⃗ := (A1 , . . . , AN ) ⊆ A(⃗ p)⃗ ⇔ Ai ⊆ A+i (p)⃗ for any i ∈ ℐ ∪ {0},

(4.16)

where A0 := X − ∑i∈ℐ Ai . Definition 4.3.1. The vector p⃗ := {p1 , . . . , pN } ∈ ℝN is an equilibrium price vector with ⃗ respect to m⃗ if there exists A⃗ ∈ 𝒪𝒮𝒫 N{m}⃗ such that A⃗ ⊆ A(⃗ p). N N ⃗ Conversely, if A ∈ 𝒪𝒮𝒫 {m}⃗ and p⃗ ∈ ℝ satisfies (4.16), then A⃗ is a competitive ⃗ m-subpartition with respect to p.⃗ An easy consequence is the following. Proposition 4.2. If p⃗ ∈ ℝN is an equilibrium price vector with respect to m,⃗ then the corresponding subpartition in 𝒪𝒮𝒫 N{m}⃗ is stable. ⃗ Let p⃗ = (p1 , . . . , pk ) be Proof. Let (i1 , . . . , ik ) be a k-chain in ℐ ∪ {0}, and xij ∈ Aij ⊆ A+ij (p). an equilibrium vector and set p0 = 0. We may assume ij ≠ ij−1 . Then by the definition of A+i (p)⃗ (see equations (4.14), (4.15)), θij (xij+1 ) − pij ≤ [θij (xij+1 ) − pij ]+ ≤ [θij+1 (xij+1 ) − pij+1 ]+ = θij+1 (xij+1 ) − pij+1 , while (recall θ0 ≡ 0) if ij = 0, θij (xij+1 ) − pij = θij (xij ) − pij = 0. Then k

k

j=1

j=1

∑ θij+1 (xij+1 ) − θij (xij+1 ) ≡ ∑[θij+1 (xij+1 ) − pij+1 ] − [θij (xij+1 ) − pij ] ≥ 0, and hence we have (4.1).

4.4 The big brother meets the free market Suppose the price vector is p⃗ ∈ ℝN . The profit of client x is ξ + (p,⃗ x) (equation (4.13)). The overall profit of the client population is Ξθ,+ (p)⃗ := ∫ ξ + (p,⃗ x)μ(dx).

(4.17)

X

Given the capacity vector m,⃗ suppose that the clients are grouped into a feasible par⃗ where m⃗ is either S or US. The total profit of tition A⃗ ∈ 𝒪𝒮𝒫 NK , where, e. g., K := {m}, ⃗ the client population is θ(A) as defined in (4.10). ⃗ The first result we state is that there is, indeed, Can we compare θ(A)⃗ to Ξθ,+ (p)? such a comparison.

44 | 4 Monge partitions Proposition 4.3. For any given A⃗ ∈ 𝒪𝒮𝒫 N{m}⃗ and p⃗ ∈ ℝN , θ(A)⃗ ≤ Ξθ,+ (p)⃗ + p⃗ ⋅ m.⃗

(4.18)

Proof. By the definition of ξ + (p,⃗ x) (equation (4.13)), θi (x) ≤ ξ + (p,⃗ x) + pi .

(4.19)

Integrate (4.19) with respect to μ over X and sum over ℐ to obtain θ(A)⃗ ≤ ∑ ∫ [ξ + (p,⃗ x) + pi ] μ(dx) = i∈ℐ A

ξ + (p,⃗ x)dμ + ∑ pi μ(Ai ) ≤ Ξθ,+ (p)⃗ + p⃗ ⋅ m,⃗

∫

i∈ℐ

⋃i∈ℐ Ai

i

(4.20)

where we used ⋃i∈ℐ Ai ⊂ X, (4.17), and A⃗ ∈ 𝒪𝒮𝒫 N{m}⃗ . It follows that equality in (4.18) at A⃗ = A⃗ 0 , p⃗ = p⃗ 0 implies that A⃗ 0 is a maximizer of θ in 𝒪𝒮𝒫 N{m}⃗ and p⃗ 0 is a minimizer of p⃗ 󳨃→ Ξθ,+ (p)⃗ + p⃗ ⋅ m⃗ in ℝN . Moreover, we have the following proposition. Proposition 4.4. There is an equality in (4.18) at (A,⃗ p)⃗ = (A⃗ 0 , p⃗ 0 ) if and only if p⃗ 0 is an equilibrium price vector with respect to m.⃗ Proof. If there is an equality in (4.18), then the inequalities in (4.20) turn into equalities as well. In particular, θ(A⃗ 0 ) ≡ ∑ ∫ θi dμi = ∑ ∫ [ξ + (p⃗ 0 , x) + p0,i ] μ(dx). i∈ℐ A

0,i

i∈ℐ A

(4.21)

0,i

But θi (x) ≤ ξ + (p⃗ 0 , x) + p0,i

for any x ∈ X and θi (x) = ξ + (p⃗ 0 , x) + p0,i iff x ∈ A+i (p⃗ 0 ) (4.22)

by definition. Hence A0,i ⊆ A+i (p⃗ 0 ). In particular, p⃗ 0 is an equilibrium price vector corresponding to the subpartition A⃗ 0 ∈ 𝒪𝒮𝒫 N{m}⃗ . Conversely, suppose p⃗ 0 is an equilibrium price vector with respect to m.⃗ Let A⃗ 0 ∈ + 𝒪𝒮𝒫 N ⃗ be the corresponding open subpartition. Then ξ (p⃗ 0 , x) + p0,i = θi (x) for any {m} + x ∈ A0,i , and (4.21) follows. Since μ(A0,i ) = mi and ξ (p⃗ 0 , x) = 0 on A0,0 ⊂ A+0 (p⃗ 0 ), we obtain that the second inequality in (4.20) is an equality as well. Given a convex K ⊂ ℝN+ , the support function of K is HK : ℝN → ℝ given by HK (p)⃗ := max p⃗ ⋅ m.⃗ ⃗ m∈K

(4.23)

In particular, if K = Km (equation (4.4)), then HK (p)⃗ ≡ ∑ mi [pi ]+ . i∈ℐ

Proposition 4.3 and (4.23) imply the following.

(4.24)

4.4 The big brother meets the free market | 45

Proposition 4.5. For any given K ⊂ ℝN+ , A⃗ ∈ 𝒪𝒮𝒫 NK , and p⃗ ∈ ℝN , ⃗ θ(A)⃗ ≤ Ξθ,+ (p)⃗ + HK (p).

(4.25)

In addition, we have the following. ⃗ then p⃗ is an equilibrium price Proposition 4.6. If there is equality in (4.25) at (A,⃗ p), ⃗ while A⃗ is a maximizer of θ vector with respect to some m⃗ 0 ∈ K verifying p⃗ ⋅ m⃗ 0 = HK (p), in 𝒪𝒮𝒫 N{m⃗ } . If, in addition, θi are non-negative and K is as given by (4.4), then: 0 i) If m⃗ ∈ ℝN++ is either S or US, then p⃗ ∈ ℝN+ . ii) If m⃗ is OS and pi > 0, then μ(Ai ) ≡ m0,i = mi , while if pi < 0, then μ(Ai ) ≡ m0,i = 0. In particular, if 0 < m0,i < mi , then pi = 0. iii) In any of the above cases, if p⃗ ∈ ℝN satisfies the equality in (4.25) for some A,⃗ then [p]⃗ + := ([p1 ]+ , . . . , [pN ]+ ) ∈ ℝN+ and A⃗ satisfies the equality in (4.25) as well. Let us linger a little bit about the meaning of (i), (ii). In the S and US cases the market is in favor of the agents. In that case no agent will offer a bonus (Remark 4.3.1) at equilibrium. In the OS case the market is in favor of the consumers, so some agents may be tempted to offer bonuses to attract clients. However, these unfortunate agents will have no clients (μ(Ai ) = 0)! If an agent i requests a positive price pi > 0 at equilibrium, it means that she is fully booked (μ(Ai ) = mi ). All other agents neither offer a bonus nor charge a price for their service (pi = 0). Finally, if the unfortunate agent i offers a bonus (pi < 0) and nevertheless gets no clients, she can obtain the same by giving her service for free pi = 0 (since she gets no profit anyway). Proof. Since now A⃗ ∈ 𝒪𝒮𝒫 NK we obtain, as in (4.20), θ(A)⃗ ≤ ∑ ∫ [ξ + (p,⃗ x) + pi ] μ(dx) = i∈ℐ A

∫

⃗ ξ + (p,⃗ x)μ(dx) + ∑ pi μ(Ai ) ≤ Ξθ,+ (p)⃗ + HK (p), i∈ℐ

⋃i∈ℐ Ai

i

(4.26)

where we used ⋃i∈ℐ Ai ⊂ X (hence Ξθ,+ (p)⃗ ≥ ∫⋃

ξ + (p,⃗ x)μ(dx)) and m0,i := μ(Ai ) ≤ mi (hence HK (p)⃗ ≥ ∑i∈ℐ pi μ(Ai )). Under the assumption θ(A)⃗ = Ξθ,+ (p)⃗ + HK (p)⃗ we obtain both i∈ℐ

∑ pi μ(Ai ) = HK (p)⃗ and

i∈ℐ

Ai

Ξθ,+ (p)⃗ =

∫

ξ + (p,⃗ x)μ(dx).

(4.27)

⋃i∈ℐ Ai

⃗ Moreover, (4.24) implies that pi ≥ 0 if M0,i > 0. If m⃗ is S or In particular, p⃗ ⋅ m⃗ 0 = HK (p). US, then μ(Ai ) = mi = mi,0 by Remark 4.2.1. Thus, m⃗ ∈ ℝN++ implies pi ≥ 0 for any i ∈ ℐ . To prove part (iii), note that HK ([p]⃗ + ) = HK (p)⃗ by (4.24), while Ξθ,+ ([p]⃗ + ) ≤ Ξ0ζ (p)⃗ by definition. Hence, the right side of (4.25) is not increasing by replacing p⃗ with [p]⃗ + . Since we assumed that p⃗ satisfies the equality at (4.25) with a given A,⃗ it implies that the same equality is satisfied for [p]⃗ + and, in particular, Ξθ,+ (p)⃗ = Ξθ,+ ([p]⃗ + ).

46 | 4 Monge partitions

4.5 All the ways lead to stable subpartitions Propositions 4.1 and 4.2 demonstrate two ways to test conditions for the stability of a given subpartition A⃗ ∈ 𝒪𝒮𝒫 N{m}⃗ . The first is by showing that A⃗ maximizes θ over 𝒪𝒮𝒫 N ⃗ , and the second is by finding an equilibrium price vector p⃗ corresponding {m} ⃗ to A. It turns out that, in fact, any stable subpartition in 𝒪𝒮𝒫 N{m}⃗ is a maximizer of θ, and admits an equilibrium price vector. Theorem 4.1. Let A⃗ ∈ 𝒪𝒮𝒫 N{m}⃗ . The following conditions are equivalent: i) A⃗ is a stable partition, ii) there exists p⃗ ∈ ℝN for which

θ(A)⃗ = Ξθ,+ (p)⃗ + p⃗ ⋅ m,⃗

(4.28)

iii) A⃗ is a maximizer of θ in 𝒪𝒮𝒫 N{m}⃗ , iv) p⃗ is a minimizer of p⃗ 󳨃→ Ξθ,+ (p)⃗ + p⃗ ⋅ m⃗ in ℝN , and A⃗ is the corresponding competitive subpartition. Proof. We already know that (ii), (iii), and (iv) are equivalent by Proposition 4.4. This and Proposition 4.2 guarantee that (ii), (iii), and (iv) imply (i) as well. Suppose (i). Let k−1

pi := sup ( ∑ θil+1 (xil ) − θil (xil )) − θik (xik ) + θi (xik ), l=1

(4.29)

where the supremum is taken over all k + 1 chains (i0 , i1 , . . . , ik ) in ℐ ∪ {0}, k ∈ ℕ ∪ {0}, satisfying i0 = 0 and xil ∈ Ail . Note that, by cyclic subadditivity, p0 ≤ 0. In fact, p0 = 0 (why?). Let i ∈ ℐ . Let α < pi and consider a k-chain realizing k−1

α < ( ∑ θil+1 (xil ) − θil (xil )) − θik (xik ) + θi (xik ). l=1

By cyclic monotonicity (inequality (4.1)) α − θi (xik ) + θ1 (xik ) ≤ 0, in particular, pi < ∞. Hence, for any j ∈ ℐ ∪ {0} and y ∈ Ai , k−1

α − θi (y) + θj (y) < ( ∑ θil+1 (xil ) − θil (xil )) − θik (xik ) + θi (xik ) − θi (y) + θj (y) ≤ pj , (4.30) l=1

4.6 Weak definition of partitions | 47

where the last inequality follows by the substitution of the (k + 1)-cycle (i1 , i2 , . . . , ik , ik+1 = i) and xik+1 = y in (4.29). Since α is any number smaller than pi , it follows that θi (y) − pi ≥ θj (y) − pj for any y ∈ Ai . Taking j = 0 we obtain, in particular, θi (y) − pi ≥ 0 for any i ∈ ℐ and y ∈ Ai . Hence [θi (y) − pi ]+ ≥ [θj (y) − pj ]+ ⃗ so p⃗ is an equilibrium price vector for any i, j ∈ ℐ ∪ {0} and y ∈ Ai , so A⃗ ⊆ A(⃗ p), (Definition 4.3.1). The result follows now from Proposition 4.4.

4.6 Weak definition of partitions Theorem 4.1(iii) shows a direct way to obtain a stable open subpartition in 𝒪𝒮𝒫 N{m}⃗ : Find such a subpartition which maximizes θ in this set. But how can we do this? Suppose we find a sequence of open subpartitions A⃗ n ∈ 𝒪𝒮𝒫 N ⃗ such that {m} lim θ(A⃗ n ) =

n→∞

sup

N ⃗ A∈𝒪𝒮𝒫 ⃗ {m}

⃗ θ(A)⃗ ≡ Σθ (m).

(4.31)

Can we identify an open subpartition A⃗ which, in some sense, is the “limit” of some ⃗ subsequence of A⃗ n ? And, if we could, can we show that θ(A)⃗ = Σθ (m)? In order to proceed, we need to assign some topology on 𝒪𝒮𝒫 N{m}⃗ . Suppose we had some metric on X. It induces a natural metric on the set of subsets of X, namely, the Hausdorff distance between A1 , A2 ⊂ X: dH (A1 , A2 ) := {sup inf d(x, y)} ∨ {sup inf d(x, y)}. x∈A1 y∈A2

x∈A2 y∈A1

Surely, the Hausdorff distance can be applied to the set of subpartitions in 𝒪𝒮𝒫 N{m}⃗ componentwise, and provides us with a metric on this set. However, the Hausdorff distance does not respect the measure μ. In particular, if An → A in the Hausdorff metric and μ(An ) = m, then μ(A) ≠ m, in general. Thus, 𝒪𝒮𝒫 N{m}⃗ is not a complete metric space under the Hausdorff distance. To overcome this difficulty, let us consider the following definition of convergence. Definition 4.6.1. A sequence of measurable sets An is said to converge weakly* to a measurable set A (An ⇀ A) if, for any continuous function ϕ on X, lim ∫ ϕdμ = ∫ ϕdμ.

n→∞

An

A

48 | 4 Monge partitions In particular, letting ϕ ≡ 1 we obtain that An ⇀ A implies limn→∞ μ(An ) = μ(A). Using this definition for each component of a partition we easily obtain the continuity of the function θ : A⃗ → ℝ with respect to the weak* convergence. What we may miss is, however, the compactness of this topology on measurable sets. Indeed, the “space” of Borel sets is not even closed under weak* convergence. Example 4.6.1. Let X = [0, 1], let μ be the Lebesgue measure, and let An := {x; ∃k even, x ∈ [k/n, (k + 1)/n)}. Then 1

1 lim ∫ ϕdx = ∫ ϕdx, n→∞ 2 0

An

but there is no set A ∈ ℬ for which ∫A ϕdx =

1 1 ∫ ϕdx 2 0

for any continuous ϕ.

Let us represent a subset A ∈ ℬ by the measure 1A dμ, where 1A is the characteristic function 1A (x) := {

1 0

if x ∈ A, otherwise.

Stated differently, we may define the set A by its action as a linear functional on the space of continuous functions on X C(X); ϕ ∈ C(X) → ∫ ϕdμ = ∫ ϕ1A dμ ∈ ℝ. A

X

We may now extend the “space” of Borel sets ℬ to the space of all bounded Borel measures on (X, ℬ), considered as linear functionals on C(X), ϕ ∈ C(X) → ∫ ϕdν ∈ ℝ, X

and define the weak* convergence of a sequence of Borel measures νn to ν by νn ⇀ ν ⇔ lim ∫ ϕdνn = ∫ ϕdν ∀ϕ ∈ C(X). n→∞

X

X

What did we gain with this notion of convergence? It turns out that the set of bounded Borel measures is closed under this notion of convergence. Moreover, it is also locally compact. In particular, we have the following. If {νn } is a sequence of Borel measures bounded by μ, then there exists a Borel measure ν ≤ μ and a subsequence {νnk } ⊂ {νn } such that limk→∞ νnk = ν in the sense of weak* convergence.

4.6 Weak definition of partitions | 49

This local compactness of the set of bounded Borel measures under weak* convergence is the key for the Kantorovich relaxation, which is the idea behind the notion of weak partitions defined in the next section. The notion of convergence of measures in general, and weak* convergence in particular, is a deep subject, but this result of the local compactness is all we need to know in order to proceed in this book. A detailed study of measure convergence can be found in [6] (and many other sources). For the convenience of the reader we expand on this subject in Appendix B.

4.6.1 Kantorovich relaxation of (sub)partitions Definition 4.6.2. A weak subpartition of (X, μ) of order N is given by N non-negative Radon measures μi on (X, ℬ), i ∈ ℐ which satisfy μ⃗ := (μ1 , . . . , μN ),

|μ|⃗ := ∑ μi ≤ μ; i∈ℐ

(4.32)

𝒮𝒫 w is the collection of all such weak partitions of (X, μ).

If there is equality in (4.32), μ⃗ is called a weak partition. The set of weak partitions is denoted 𝒫 w ⊂ 𝒮𝒫 w . ℝN+ ,

Motivated by the above, we generalize (4.2) as follows: For any m⃗ = (m1 , . . . , mN ) ∈ w

w

𝒮𝒫 ≤m⃗ := {μ⃗ := (μ1 , . . . , μN ) ∈ 𝒮𝒫 ; μi (X) ≤ mi , i ∈ ℐ } ,

(4.33)

and, more generally, w

w

⃗ 𝒮𝒫 K := {μ⃗ := (μ1 , . . . , μN ) ∈ 𝒮𝒫 ; μ(X) ∈ K}

(4.34)

for a given closed set K ⊂ ℝN . In addition, we extend the function θ (4.9) to 𝒮𝒫 w as θ(μ)⃗ := ∑ ∫ θi (x)μi (dx). i∈ℐ X

(4.35)

Remark 4.6.1. In this and the next chapter we do not need to assume the condition that μ is an atomless measure declared in Standing Assumption 4.1.1. In particular, we may even assume that X is a finite discrete set. Example 4.6.2. Let X be a finite set {x1 , . . . , xn }, and let 𝒥 := {1, . . . , n}. Let n

μ := ∑ αj δxj , j=1

(4.36)

50 | 4 Monge partitions where αj > 0, ∑nj=1 αj = 1, and δx is the Dirac measure at x ∈ X. A weak partition is given by n

μi := ∑ π(i, j)δxj , j=1

i ∈ ℐ,

(4.37)

where ∑Ni=1 π(i, j) = αj and π(i, j) ≥ 0 for any j ∈ 𝒥 , i ∈ ℐ . Under the same setting we may present θ⃗ on X in terms of N × n matrix {θij }. Hence ⃗ θ(μ) takes the form N

n

θ(μ)⃗ := ∑ ∑ θij π(i, j). i=1 j=1

What is the point behind such a generalization? Recall (4.31). If we could prove the existence of a maximizer A⃗ for θ in 𝒪𝒮𝒫 N{m}⃗ , we would have a stable (sub)partition in our hand! The problem is that we do not have the tool to prove the existence of such a maximizer in the set of open (sub)partitions 𝒪𝒮𝒫 N{m}⃗ . However, we can obtain, quite cheaply, the existence of a maximizer for θ on sets of weak (sub)partitions. Here we take advantage of the local compactness of the space of Borel measures with respect to the weak* topology, and apply it componentwise to the weak partitions: μ⃗ n := (μ1n , . . . , μNn ) ⇀ μ := (μ1 , . . . , μN ) ⇔ μin ⇀ μi ,

∀ i ∈ ℐ.

Thus w a) 𝒮𝒫 w K is a compact subset of 𝒮𝒫 , w b) θ is continuous on 𝒮𝒫 K . This means that for any converging sequence μ⃗ n ⇀ μ⃗ in 𝒮𝒫 w K, ⃗ lim θ(μ⃗ n ) = θ(μ).

n→∞

Indeed, let us consider a maximizing sequence μ⃗ n satisfying ⃗ lim θ(μ⃗ n ) = Σ̄ θ (K) := sup θ(μ).

n→∞

w ⃗ μ∈𝒮𝒫 K

(4.38)

By (a) we get the existence of a subsequence μ⃗ k ⇀ μ⃗ ∈ 𝒮𝒫 w K , and from (b) we obtain ⃗ = θ(μ)⃗ and μ⃗ is a maximizer! ⃗ Hence Σ̄ Kθ (m) Σ̄ θ (K) = θ(μ). The continuity (b) of θ on 𝒮𝒫 w K follows from Assumption 4.0.1(ii). In particular, we have the following. Theorem 4.2. For any closed set K ⊂ ℝN there exists a maximizer μ⃗ of θ (4.35) in 𝒮𝒫 w K. So, to answer question 4.7(1) affirmatively we just have to prove that this maximizer is, in fact, in 𝒪𝒮𝒫 Nm⃗ . Such a result will guarantee, in particular, that Σ̄ θ (K) defined in (4.38) is equal to Σθ (K) defined in (4.11).

4.6 Weak definition of partitions | 51

4.6.2 Birkhoff theorem In the context of Example 4.6.2 the sets 𝒫{wm}⃗ can certainly be empty. Consider the particular case of an empirical atomic measure N

μ := ∑ δxi ,

and m⃗ := (1, . . . , 1).

i=1

(4.39)

In that case we observe that we can embed any atomic weak partition (4.37) in the set of N × N doubly stochastic matrices N

N

i=1

j=1

Π := {αi,j ≥ 0; ∑ αi,j = ∑ αi,j = 1}.

(4.40)

A bijection τ : ℐ ↔ {x1 , . . . , xN } corresponds to a matrix πτ in the set of permutation matrices 𝒫 : τ ⇒ πτ (i, j) := (

1 0

if j = τ(i) ) ∈ 𝒫 ⊂ Π. if j ≠ τ(i)

Now we compare (4.9) with θi (xj ) := θij and mi = 1 with (2.18). Let N N

θ(π) := ∑ ∑ θij π(i, j). i=1 j=1

(4.41)

Then we obtain immediately that N

̂ θ(τ) := ∑ θiτ(i) = θ(πτ ). i=1

In particular, ̂ max θ(π) ≥ max θ(τ). τ∈𝒫

π∈Π

(4.42)

It is, however, somewhat surprising that there is, in fact, an equality in (4.42). This follows from the following theorem of Birkhoff. Theorem 4.3. The set Π is the convex hull2 of 𝒫 . Namely, for any π ∈ Π there exists a set of permutations {τj } ⊂ 𝒫 and positive numbers {aj } satisfying ∑j aj = 1 such that ∑ aj πτj = π. j

2 See Appendix A.1.

52 | 4 Monge partitions Birkhoff’s theorem implies the following. Proposition 4.7. There is an equality in (4.42). From Theorem 4.2 applied to the case of Example 4.6.2 we obtain that a maximizer π̄ of (4.41) is, in fact, a stable weak partition N

N

j=1

j=1

̄ ̄ j)δxj , . . . , ∑ π(N, j)δxj ) μ⃗ := (∑ π(1, of (4.39). From the equality in (4.42) due to the Birkhoff theorem we get that π̄ is also a permutation matrix which maximizes the right side of (4.42) as well. Then this permutation is the Monge solution τ in the sense of Definition 2.6.1. There are several proofs (mostly algebraic) of the Birkhoff theorem in the literature. The following elegant proof of Zhu is based on a variational argument, in the spirit of this book. Here is a sketch of the argument of Zho. Let π ∈ Π be a doubly stochastic matrix. Define the function f on the set 𝒢 of real-valued N × N matrices: f (G) := ln ( ∑ eG:(π−τ) ) ,

G ∈ 𝒢.

τ∈𝒫

The function f is differentiable and its derivative at G ∈ 𝒢 is f 󸀠 (G) = ∑ λG (σ)(π − σ), σ∈𝒫

where λG (σ) :=

eG:(π−σ) ∑τ∈𝒫 eG:(π−τ)

satisfies ∑σ∈𝒫 λG (σ) = 1. Now, assume we know that there exists a minimizer G0 ∈ 𝒢 of f . Then f 󸀠 (G0 ) = 0, namely, ∑ λG0 (σ)(π − σ) = 0 󳨐⇒ π = ∑ λG0 (σ)σ,

σ∈𝒫

σ∈𝒫

which implies Birkhoff’s theorem. However, the assumption that there exists a minimizer of f of 𝒢 is too strong. To justify this argument, Zho first claimed that f is bounded from below on 𝒢 , and then applied a very useful and elementary lemma: Approximate Fermat principle: If f is differentiable on the entire space and bounded from below, then for any ϵ > 0 there exists an approximate critical point G for which |f 󸀠 (G)| ≤ ϵ. This, and the local compactness of 𝒢 , is enough to complete the argument. The fact that f is bounded from below follows from another elementary argument of Zho which implies that for any G ∈ 𝒢 and π ∈ Π there exists τ ∈ 𝒫 for which G : (π − τ) ≥ 0.

4.7 Summery and beyond | 53

4.7 Summery and beyond What did we learn so far? If m⃗ ∈ ℝN+ is S or US, then: i) By Proposition 4.3, sup

N ⃗ A∈𝒪𝒮𝒫 ⃗ {m}

⃗ ≤ inf Ξθ,+ (p)⃗ + p⃗ ⋅ m.⃗ θ(A)⃗ := Σθ⃗ (m) ⃗ N p∈ℝ

(4.43)

ii) From Proposition 4.3 and Theorem 4.1: The equality in (4.43) is a necessary condition for the existence of a stable subpartition in 𝒪𝒮𝒫 N{m}⃗ . iii) From Theorem 4.2: There exists a maximizer on the set of weak subpartitions max θ(μ)⃗ ≥

w ⃗ μ∈𝒮𝒫 m

sup

N ⃗ A∈𝒪𝒮𝒫 ⃗ {m}

⃗ θ(A).

The questions still left open, at this stage, are: 1. Is there a maximizer on the left side of (4.43)? 2. If so, is this maximizer unique in 𝒪𝒮𝒫 N{m}⃗ ? 3. What about a minimizer p⃗ ∈ ℝN of the right side of (4.43)? 4. The same questions regarding the maximizer in 𝒪𝒮𝒫 NK and the minimizer in ℝN of sup

N ⃗ A∈𝒪𝒮𝒫 K

for a given K ⊂ ℝN+ .

⃗ ≤ inf Ξθ,+ (p)⃗ + HK (p)⃗ θ(A)⃗ := ΣKθ⃗ (m) ⃗ N p∈ℝ

(4.44)

|

Part II: Multipartitions

5 Weak multipartitions There are many ways of going forward, but only one way of standing still. (F. D. R.)

5.1 Multipartitions Let us now generalize the definition of partitions and capacity (Section 4.2) in a natural way. Suppose the there is a set of goods 𝒥 = {1, . . . , J}. Each customer x ∈ X consumes a given fraction ζj (x) ≥ 0 of j ∈ 𝒥 . The consumption vector ζ ̄ := (ζ1 (x), . . . , ζJ (x)) is defined such that ∑Jj=1 ζj (x) = 1, thus ζ ̄ (x) ∈ ΔJ (1) for any x ∈ X. We further assume ζ ̄ ∈ C(X; ΔJ (1)),

̄ thus μ(dx) := ζ ̄ (x)μ(dx) ∈ ℳN+ (X).

(5.1)

Each agent i ∈ ℐ can supply each of the goods j ∈ 𝒥 under a prescribed capacity. Here (j) mi ≥ 0 is the capacity of agent i for goods j and (J) J m⃗ i := (m(1) i , . . . , mi ) ∈ ℝ+ ,

i ∈ ℐ.

(j) Definition 5.1.1. The capacity matrix {mi } := M⃗ := {m⃗ 1 , . . . , m⃗ N } is an N × J matrix of positive entries. The set of all such matrices is denoted 𝕄+ (N, J).

For an admissible weak (sub)partition μ⃗ := (μ1 , . . . , μN ) ∈ 𝒮𝒫 w , each agent should be able to supply the part μi of the population. This implies μi (ζj ) := ∫ ζj (x)μi (dx) ≤ mi ; (j)

(i, j) ∈ ℐ × 𝒥 .

X

Let now 𝒦 ⊂ 𝕄+ (N, J). We generalize (4.33) for the set of weak subpartitions w,ζ ̄

w

𝒮𝒫 𝒦 := {μ⃗ := (μ1 , . . . , μN ) ∈ 𝒮𝒫 ; μ(⃗ ζ ̄ ) ∈ 𝒦} .

(5.2)

⃗ is a singleton, then we denote the corresponding subpartitions If 𝒦 = {M}

by 𝒮𝒫

w,ζ ̄ . ⃗ {M}

w,ζ

w,ζ ̄

The set of partitions satisfying ∑i μi = μ is denoted by 𝒫 ⃗ ⊂ 𝒮𝒫 ⃗ . {M} {M} Let M⃗ ζ (μ)⃗ := {μi (ζj )} ∈ 𝕄+ (N, J).

(5.3)

The conditions on M⃗ ∈ 𝕄+ (N, J) for which the corresponding subpartition sets are not empty (feasibility conditions) are not as simple as (4.5)–(4.7). In particular, the notions of saturation (S), undersaturation (US), and oversaturation (OS) presented in Section 4.2 ((4.7), (4.6), and (4.5)) should be generalized. https://doi.org/10.1515/9783110635485-005

58 | 5 Weak multipartitions Definition 5.1.2. The feasibility sets with respect to μ̄ := ζ ̄ μ are (equation (5.3)) ⃗ μ⃗ ∈ 𝒫 w }, ΔN (μ)̄ := {M⃗ = M⃗ ζ (μ);

⃗ μ⃗ ∈ 𝒮𝒫 w }. ΔN (μ)̄ := {M⃗ = M⃗ ζ (μ);

Equivalently, w,ζ ΔN (μ)̄ := {M;⃗ 𝒫 ⃗ ≠ 0},

w,ζ ΔN (μ)̄ := {M;⃗ 𝒮𝒫 ⃗ ≠ 0}.

{M}

{M}

Remark 5.1.1. Note that for any M⃗ ∈ ΔN (μ)̄ ∑ mi ≡ ∑ μi (ζj ) = μi (X),

(5.4)

∑ mi ≡ ∑ μi (ζj ) ≤ μ(ζj ).

(5.5)

(j)

j∈𝒥

j∈𝒥

while (j)

i∈ℐ

i∈ℐ

By definition ∑

N

mi = ∑ μi ( ∑ ζj ) = ∑ μi (X). (j)

i∈ℐ,j∈𝒥

i∈ℐ

j∈𝒥

i=1

If M⃗ ∈ ΔN (μ)̄ (so ∑i∈ℐ μi = μ) we get an equality in (5.5): N

∑ mi = μ(ζj ).

(5.6)

(j)

i=1

Example 5.1.1. If J = 1, then ΔN is the simplex of all N-vectors N

ΔN = ΔN (1) := {m⃗ := (m1 , . . . , mN ) ∈ ℝN+ ; ∑ mi = 1} i=1

and ΔN (μ)̄ is the subsimplex ΔN = ΔN (1) := {m⃗ := (m1 , . . . , mN ) ∈ ℝN+ ; ∑ mi ≤ 1} . i∈ℐ

The natural generalizations of oversaturation (4.5), saturation (4.6), and undersaturation (4.7) are as follows: ̄ oversaturated (OS): if M⃗ ∈ 𝕄+ (N, J) − ΔN (μ),

(5.7)

̄ saturated (S): if M⃗ ∈ ΔN (μ),

(5.8)

̄ undersaturated (US): if M⃗ ∈ ΔN (μ)̄ − ΔN (μ).

(5.9)

Proposition 5.1. The sets ΔN (μ)̄ ⊂ ΔN (μ)̄ are compact, bounded, and convex in 𝕄+ (N, J). The set ΔN (μ)̄ has a non-empty interior in 𝕄+ (N, J). Consider Example 5.1.1 again. Here ΔN (μ)̄ is an (N − 1)-dimensional simplex in ℝN , so it has an empty interior. It is not a priori clear that the interior of ΔN (μ)̄ is not empty in the general case. However, it is the case by Corollary 5.2.2 below.

5.2 Feasibility conditions | 59

5.2 Feasibility conditions 5.2.1 Dual representation of weak (sub)partitions Here we attempt to characterize the feasibility sets by a dual formulation. For this we return to the “market” interpretation of Chapter 4.3. Definition 5.2.1. Let P⃗ be an N ×J matrix of real entries. Any such matrix is represented by its rows: P⃗ = (p⃗ 1 , . . . , p⃗ N ), where p⃗ i ∈ ℝJ . The set of all these matrices is denoted by 𝕄󸀠 (N, J). Here 𝕄+ (N, J) is as given in Definition 5.1.1 and 𝕄󸀠 (N, J) are considered as dual spaces, under the natural duality action N

J

(j) (j) P⃗ : M⃗ := ∑ p⃗ i ⋅ m⃗ i ≡ ∑ ∑ pi mi , i∈ℐ

i=1 j=1

where p⃗ ⋅ m⃗ is the canonical inner product in ℝJ . Let ζ ̄ := (ζ1 , . . . , ζJ ) : X → ℝJ+ verify the assumption (5.1). Define, for x ∈ X and P⃗ = (p⃗ 1 , . . . , p⃗ N ), ξζ0 (x, P)⃗ := max p⃗ i ⋅ ζ ̄ (x) : X × 𝕄󸀠 (N, J) → ℝ, ξζ0,+ (x, P)⃗ :=

i∈ℐ ξζ0 (x, P)⃗

∨ 0 : X × 𝕄󸀠 (N, J) → ℝ+ ,

⃗ : 𝕄󸀠 (N, J) → ℝ, Ξ0ζ (P)⃗ := μ(ξζ0 (⋅, P)) 0,+ 󸀠 ⃗ ⃗ Ξ0,+ ζ (P) := μ(ξζ (⋅, P)) : 𝕄 (N, J) → ℝ+ .

(5.10) (5.11) (5.12) (5.13)

Here we see a neat, equivalent definition of the feasibility sets (5.1.2). ̄ if and only if Theorem 5.1. M⃗ ∈ ΔN (μ)̄ (resp. M⃗ ∈ ΔN (μ)) a) Ξ0ζ (P)⃗ − P⃗ : M⃗ ≥ 0;

⃗ ⃗ ⃗ resp. b) Ξ0,+ ζ (P) − P : M ≥ 0

(5.14)

for any P⃗ ∈ 𝕄󸀠 (N, J). Moreover, Ξ0ζ and Ξ0,+ are the support functions1 of ΔN (μ)̄ and ζ ̄ respectively: ΔN (μ), ⃗ sup P⃗ : M⃗ = Ξ0ζ (P);

⃗ N (μ)̄ M∈Δ

holds for any P⃗ ∈ 𝕄󸀠 (N, J). 1 See Appendix A.6.

⃗ sup P⃗ : M⃗ = Ξ0,+ ζ (P)

⃗ ̄ M∈Δ N (μ)

(5.15)

60 | 5 Weak multipartitions Recall Definitions A.3.1 and A.6.2 in Appendix A.2. Lemma 5.1. The functions Ξ0ζ and Ξ0,+ are convex, continuous, and positively homogeζ neous of order 1 functions on 𝕄󸀠 (N, J).

Proof. By Proposition A.7 and (5.10), (5.11) we obtain that P⃗ 󳨃→ ξζ0 and P⃗ 󳨃→ ξζ0,+ are

convex (as functions of 𝕄󸀠 (N, J)) for any x ∈ X. Indeed, they are maximizers of linear (affine) functions on 𝕄󸀠 (N, J). Hence Ξ0ζ , Ξ0,+ are convex on 𝕄󸀠 (N, J) as well from defiζ nition (5.12), (5.13). Since X is compact, Ξ0 , Ξ0,+ are finite-valued for any P⃗ ∈ 𝕄󸀠 (N, J) so ζ

ζ

the essential domains of both coincide with 𝕄󸀠 (N, J). Hence both functions are continuous by Proposition A.6. Both functions are positive homogeneous of order 1 by definition. Corollary 5.2.1. M⃗ is an inner point of ΔN (μ)̄ if and only if P⃗ = 0 is the only case where (5.14b) holds with an equality. ⃗ P⃗ : M⃗ is positively homogeneous by Lemma 5.1, it (P)− Proof of the corollary. Since Ξ0,+ ζ ̄ If it is a strict minimizer, follows that P⃗ = 0 is a minimizer of (5.14b) for any M⃗ ∈ ΔN (μ). 0,+ ⃗ 0,+ ⃗ ⃗ ⃗ then Ξ (P) − P : M > 0 for any P ≠ 0. Since Ξ is continuous at any point in its ζ

ζ

essential domain (in particular at P⃗ = 0), there exists α > 0 such that Ξ0ζ (P)⃗ − P⃗ : M⃗ > α for any P⃗ ∈ 𝕄󸀠 (N, J),

|P|⃗ := ∑ |p⃗ i | = 1. i∈ℐ

Hence there exists an open neighborhood O of M⃗ ∈ 𝕄+ (N, J) and α󸀠 > 0 for which Ξ0,+ (P)⃗ − P⃗ : M⃗ 󸀠 > α󸀠 for any M⃗ 󸀠 ∈ O and |P|⃗ = 1. Hence Ξ0,+ (P)⃗ − P⃗ : M⃗ 󸀠 > 0 for any ζ ζ P⃗ ≠ 0. Hence M⃗ 󸀠 ∈ Δ (μ)̄ for any M⃗ 󸀠 ∈ O, and hence M⃗ is an inner point of Δ (μ)̄ by N

N

Theorem 5.1. Conversely, assume there exists P⃗ 0 ≠ 0 for which Ξ0,+ (P⃗ 0 ) − P⃗ 0 : M⃗ = 0. Then ζ 0,+ ⃗ 󸀠 󸀠 󸀠 Ξ (P0 ) − P⃗ 0 : M⃗ < 0 for any M⃗ for which P⃗ 0 : (M⃗ − M⃗ ) < 0. By Theorem 5.1 it follows ζ

̄ so M⃗ is not an inner point of ΔN (μ). ̄ that M⃗ 󸀠 ∈ ̸ ΔN (μ),

⃗ where Since p⃗ ⋅ m⃗ i = μi (p⃗ ⋅ ζ ̄ ), it follows from (5.6), (5.10), (5.12) that P⃗ : M⃗ = Ξ0ζ (P), ⃗ so Ξ0,+ (P)⃗ − P⃗ : M⃗ = 0 for ⃗ If, in addition, p⃗ ∈ ℝJ+ , then Ξ0,+ (P)⃗ = Ξ0ζ (P), P⃗ = (p,⃗ . . . , p). ζ ζ any M⃗ ∈ ΔN (μ)̄ and any such P.⃗ From Corollary 5.2.1 we obtain Corollary 5.2.2. ̄ ΔN (μ)̄ ⊂ 𝜕ΔN (μ). In particular, ΔN (μ)̄ has no interior points in 𝕄+ (N, J).

5.2 Feasibility conditions | 61

5.2.2 Proof of Theorem 5.1 ̄ then Lemma 5.2. If M⃗ ∈ ΔN (μ), Ξ0ζ (P)⃗ − P⃗ : M⃗ ≥ 0 ̄ then for any P⃗ ∈ 𝕄󸀠 (N, J). Likewise, if M⃗ ∈ ΔN (μ), ⃗ ⃗ ⃗ Ξ0,+ ζ (P) − P : M ≥ 0 for any P⃗ ∈ 𝕄󸀠 (N, J). Recalling Proposition A.12 we can formulate Lemma 5.2 as follows: The sets ΔN (μ)̄ ̄ are contained in the essential domains of the Legendre transforms of Ξ0ζ (resp. ΔN (μ)) (resp. Ξ0,+ ). ζ

̄ By definition, there exists μ⃗ ∈ 𝒮𝒫 w,ζ⃗ such that μi (ζj ) = m(j) . Proof. Assume M⃗ ∈ ΔN (μ). i ̄

{M}

Also, since ∑i∈ℐ μi ≤ μ,

0,+ 0,+ ⃗ ⃗ ⃗ Ξ0,+ ζ (P) = μ(ξζ (⋅, P)) ≥ ∑ μi (ξζ (⋅, P)), i∈ℐ

while from (5.11) ξζ0,+ (x, P)⃗ ≥ p⃗ i ⋅ ζ ̄ (x), so ̄ ⃗ ⃗ ⃗ ⃗ Ξ0,+ ζ (P) ≥ ∑ pi ⋅ μi (ζ ) = P : M. i∈ℐ

The case for Ξ0ζ is proved similarly. In order to prove the second direction of Theorem 5.1 we need the following definition of regularized maximizer. Definition 5.2.2. Let a⃗ := (a1 , . . . , aN ) ∈ ℝN . Then, for ϵ > 0, max(a)⃗ := ϵ ln (∑ eai /ϵ ) . ϵ

i∈ℐ

Lemma 5.3. For any ϵ > 0, maxϵ (⋅) is a smooth convex function on ℝN . In addition maxϵ1 (a)⃗ ≥ maxϵ2 (a)⃗ ≥ maxi∈ℐ (ai ) for any a⃗ ∈ ℝN , ϵ1 > ϵ2 > 0, and lim max(a)⃗ = max(ai ).

(5.16)

⃗ , max {−ϵ ∑ βi ln βi + β⃗ ⋅ a}

(5.17)

ϵ↘0

ϵ

i∈ℐ

Proof. Consider

β⃗

i∈ℐ

62 | 5 Weak multipartitions where the maximum is taken on the simplex β⃗ := (β1 . . . βN ),

N

βi ≥ 0,

∑ βi = 1. 1

Note that (5.17) is a strictly concave function, and its unique maximizer is eai /ϵ 0. In addition, − ∑i∈ℐ βi ln βi is maximized at βi = 1/N (show it!), so 0 ≤ − ∑i∈ℐ βi ln βi ≤ ln N. It follows that max(a)⃗ ∈ [max(ai ), max(ai ) + ϵ ln N], ϵ

i∈ℐ

i∈ℐ

and (5.16) follows. Definition 5.2.3. We have ξζϵ (x, P)⃗ := max (p⃗ 1 ⋅ ζ ̄ (x), . . . , p⃗ N ⋅ ζ ̄ (x)) : X × 𝕄󸀠 (N, J) → ℝ,

(5.18)

⃗ : 𝕄󸀠 (N, J) → ℝ. Ξϵζ (P)⃗ := μ(ξζϵ (⋅, P))

(5.19)

ϵ

Also, for each P⃗ ∈ 𝕄󸀠 (N, J) and i ∈ ℐ set μ(i P) (dx) := ⃗

ep⃗ i ⋅ζ (x)/ϵ ̄

∑k∈ℐ ep⃗ k ⋅ζ (x)/ϵ ̄

μ(dx).

(5.20)

Likewise, ξζϵ,+ (x, P)⃗ := max (p⃗ 1 ⋅ ζ ̄ (x), . . . , p⃗ N ⋅ ζ ̄ (x), 0) : X × 𝕄󸀠 (N, J) → ℝ+ , ϵ ϵ,+ ⃗ Ξζ (P)

⃗ : 𝕄󸀠 (N, J) → ℝ+ , := μ (ξζϵ,+ (⋅, P))

(5.21) (5.22)

and μ(i P,+) (dx) := ⃗

ep⃗ i ⋅ζ (x)/ϵ ̄

1 + ∑k∈ℐ ep⃗ k ⋅ζ (x)/ϵ ̄

μ(dx).

(5.23)

Since maxϵ (⋅) is smooth due to Lemma 5.3, Lemma 5.4 below follows from the above definition via an explicit differentiation. Lemma 5.4. For each ϵ > 0, Ξϵζ (resp. Ξϵ,+ ) is a convex, smooth function on 𝕄󸀠 (N, J). In ζ addition 𝜕Ξϵζ (P)⃗ (j)

𝜕pi

= μ(i P) (ζj ) ⃗

resp.

𝜕Ξϵ,+ (P)⃗ ζ (j)

𝜕pi

= μ(i P,+) (ζj ). ⃗

5.2 Feasibility conditions | 63

The proof of Theorem 5.1 is obtained from Lemma 5.5 below, whose proof is an easy exercise, using (5.17). Lemma 5.5. For any ϵ, δ > 0 and M⃗ ∈ 𝕄+ (N, J), δ P⃗ → Ξϵζ (P)⃗ + |P|⃗ 2 − P⃗ : M⃗ 2

(5.24)

is a strictly convex function on 𝕄󸀠 (N, J). In addition ⃗ Ξϵζ (P)⃗ ≥ Ξ0ζ (P),

(5.25)

so if (5.14) is satisfied, then Ξϵζ (P)⃗ − P⃗ : M⃗ ≥ 0 for any P⃗ ∈ 𝕄󸀠 (N, J). The same statement holds for Ξϵ,+ as well. ζ Proof of Theorem 5.1. Lemma 5.2 gives us the “only if” direction. From Lemma 5.5 we obtain the existence of a minimizer P⃗ ϵ,δ ∈ 𝕄󸀠 (N, J) of (5.24) for any ϵ, δ > 0, provided (5.14) holds. Moreover, from Lemma 5.4 we also get that this minimizer satisfies m⃗ i =

𝜕 ϵ (P⃗ ϵ,δ ) ̄ Ξ + δp⃗ ϵ,δ (ζ ) + δp⃗ ϵ,δ i = μi i . ϵ,δ ζ 𝜕p⃗ i

(5.26)

By convexity of Ξϵζ : Ξϵζ (0)⃗ ≥ Ξϵζ (P)⃗ − P⃗ : ∇Ξϵζ (P)⃗ for any P⃗ ∈ 𝕄󸀠 (N, J).

(5.27)

We apply p⃗ ϵ,δ to (5.26), use (5.27), and sum over i = 1, . . . , N, recalling P⃗ ϵ,δ = (p⃗ ϵ,δ 1 ,..., i ϵ,δ ⃗ p⃗ N ), M := (m⃗ 1 , . . . , m⃗ N ), obtaining 󵄨 󵄨2 󵄨 󵄨2 P⃗ ϵ,δ : ∇Ξϵζ (P⃗ ϵ,δ ) + δ 󵄨󵄨󵄨󵄨P⃗ ϵ,δ 󵄨󵄨󵄨󵄨 − P⃗ ϵ,δ : M⃗ =≥ Ξϵζ (P⃗ ϵ,δ ) − Ξϵζ (0)⃗ + δ 󵄨󵄨󵄨󵄨P⃗ ϵ,δ 󵄨󵄨󵄨󵄨 − P⃗ ϵ,δ : M.⃗ It follows from (5.14), (5.25), (5.28) that 󵄨 󵄨2 −Ξϵζ (0)⃗ + δ 󵄨󵄨󵄨󵄨P⃗ ϵ,δ 󵄨󵄨󵄨󵄨 ≤ 0, and hence 󵄨 󵄨 ⃗ δ 󵄨󵄨󵄨󵄨P⃗ ϵ,δ 󵄨󵄨󵄨󵄨 ≤ √δ√Ξϵζ (0). Hence (5.26) implies ⃗ ϵ,δ lim μ(P ) (ζ ̄ ) δ→0 i

= m⃗ i .

(5.28)

64 | 5 Weak multipartitions ⃗ ϵ,δ )

By the compactness of C ∗ (X) and since ∑i∈ℐ μ(i P subsequence δ → 0 along which the limit ⃗ ϵ,δ lim μ(P ) δ→0 i

= μ via (5.20) we can choose a

⃗ϵ)

:= μ(i P

holds for any i ∈ ℐ . It follows that ⃗ϵ

⃗ϵ

∑ μ(i P ) = μ; μ(i P ) (ζ ̄ ) = m⃗ i

i∈ℐ

w ⃗ ̄ for any i = 1, . . . , N, and hence μ⃗ ∈ 𝒫M ⃗ , so M ∈ ΔN (μ). ⃗ ̄ The proof for M ∈ ΔN (μ) is analogous. Finally, the proof of (5.15) follows from (5.14) and Proposition A.12, taking advantage of the homogeneity of Ξ0ζ (resp. the positive homogeneity of Ξ0,+ ). ζ

5.3 Dominance We now consider generalized (sub)partitions from another point of view. A stochastic N-matrix S = {ski } is an N × N matrix of non-negative entries such that N

∑ ski = 1 ∀i = 1, . . . , N.

(5.29)

k=1

We observe that if μ⃗ is a (sub)partition, then N

N

i=1

i=1

Sμ⃗ := (∑ s1i μi , . . . , ∑ sNi μi ) is a (sub)partition as well. It follows from Definition 5.1.2 and (5.3) that if M⃗ ∈ ΔN (μ)̄ ̄ then (resp. M⃗ ∈ ΔN (μ)), SM⃗ ∈ ΔN (μ)̄

̄ (resp. SM⃗ ∈ ΔN (μ)).

Here SM⃗ := (Sm⃗ (1) , . . . , Sm⃗ (J) ), where M⃗ = (m⃗ (1) , . . . , m⃗ (J) ), m⃗ (j) ∈ ℝN+ . Definition 5.3.1. Let M,⃗ M⃗ ∈ 𝕄+ (N, J). If there exists a stochastic matrix S such that ⃗ then M⃗ is said to dominate M⃗ (M⃗ ≻ M). ⃗ M⃗ = SM, Assume S is such a stochastic matrix satisfying M⃗ = SM.⃗ Let (J) J m⃗ i := (m(1) i , . . . , mi ) ∈ ℝ+ ,

J

mi := ∑ mi j=1

(j)

(5.30)

5.3 Dominance | 65

(and similarly for m⃗ i , mi ). We have N

m⃗ i = ∑ sik m⃗ k

(5.31)

k=1

(as an equality in ℝJ ). Summing the components in ℝJ of both sides of (5.31) and dividing by mi , we obtain N

∑ sik

k=1

mk = 1. mi

(5.32)

Dividing (5.31) by mi we obtain N m⃗ i m m⃗ = ∑ (sik k ) k . mi k=1 mi mk

(5.33)

The Jensen inequality and (5.32), (5.33) imply F(

N m⃗ i m m⃗ ) ≤ ∑ (sik k ) F ( k ) mi m mk i k=1

for any convex function F : ℝJ+ → ℝ. Multiplying the above by mi and summing over i = 1, . . . , N, we get, using (5.29), N

∑ mi F ( i=1

N m⃗ m⃗ i ) ≥ ∑ mi F ( i ) . mi mi i=1

(5.34)

⃗ then (5.34) holds for any convex function on ℝJ . It can be We obtained that if M⃗ ≻ M, + shown, in fact, that the reversed direction holds as well. Proposition 5.2. We have M⃗ ≻ M⃗ iff (5.34) holds for any convex F : ℝJ+ → ℝ. The definition of dominance introduced above is an extension of a definition given by H. Joe ([25, 24]). In these papers Joe introduced the notion of w-dominance on ℝN+ as ⃗ follows: For a given a vector w⃗ ∈ ℝN++ , the vector x⃗ ∈ ℝN+ is said to be w-dominant over N y⃗ ∈ ℝ+ (x⃗ ≻w y)⃗ iff there exists a stochastic matrix S preserving w⃗ and transporting x⃗ to y,⃗ i. e., Sx⃗ = y⃗

and Sw⃗ = w.⃗

Evidently, it is a special case of our definition where J = 2. The condition of x⃗ ≻w y⃗ is shown to be equivalent to N

∑ wj ψ ( j=1

yj

wj

N

) ≤ ∑ wj ψ ( j=1

xj

wj

)

(5.35)

66 | 5 Weak multipartitions for any convex function ψ : ℝ+ → ℝ. The reader should observe that (5.35) follows from (5.34) in the case J = 2 upon defining m⃗ 1 = x,⃗ m⃗ 1 = y,⃗ m⃗ 2 = λw⃗ − x,⃗ m⃗ 2 = λw⃗ − y⃗ (where λ is large enough such that both m⃗ 2 , m⃗ 2 ∈ ℝJ+ )2 and setting ψ(x) = F(s/λ, 1−s/λ). We now present a generalization of Proposition 5.2. Let ζ ̄ := (ζ1 , . . . , ζN ) : X → ℝJ+ be a measurable function, as defined in (5.1). Con(j) sider M⃗ := {mi } satisfying (5.4), (5.5). Recall (5.30). ̄ if and only if Theorem 5.2. We have M⃗ ∈ ΔN (μ)̄ (resp. M⃗ ∈ ΔN (μ)) m⃗ μ(F(ζ ̄ )) ≥ ∑ mi F ( i ) mi i∈ℐ

(5.36)

is satisfied for any convex F : ℝJ+ → ℝ (resp. F : ℝJ+ → ℝ+ ). Recall Remark 4.6.1. Note that if we choose X = ℐ := {1, . . . , N}, μ({i}) := mi , the (j) “deterministic” partition μi ({k}) := μ({i})δi,k , and mi := ζj ({i})μ({i}) then Theorem 5.2 implies Proposition 5.2. Proof. By the definition of ΔN (μ)̄ there exists a weak partition μ⃗ = (μ1 , . . . , μN ) such that (j) mi = μi (ζj ). In particular, mi = μi (X). Since F is convex we apply Jensen’s inequality μi (F(ζ ̄ )) ≥ μi (X)F (

∫ ζ ̄ dμi μi (X)

) := mi F (

m⃗ i ). mi

(5.37)

Summing over i ∈ ℐ and using μ = ∑i∈ℐ μi we obtain the result. ̄ then there exists a weak subpartition μ⃗ = (μ1 , . . . , μN ) such that If M⃗ ∈ ΔN (μ), (j) mi = μi (ζj ) and ∑Ni=1 μi ≤ μ. In that case inequality (5.36) still follows from (5.37), taking advantage of F ≥ 0. (1) (N) ̄ By Lemma 5.2 there exists P⃗ = Suppose now M⃗ := (m⃗ , . . . , m⃗ ) ∈ ̸ ΔN (μ). 󸀠 (p⃗ 1 , . . . , p⃗ N ) ∈ 𝕄 (N, J) such that Ξ0ζ (P)⃗ < P⃗ : M⃗ := ∑ p⃗ i ⋅ m⃗ i . i∈ℐ

(5.38)

Define the function F = F(ζ ̄ ) : ℝJ+ → ℝ: F(ζ ̄ ) := max p⃗ i ⋅ ζi i∈ℐ

(resp. F+ (ζ ̄ ) := max[p⃗ i ⋅ ζi ]+ ≡ F(ζ ) ∨ 0). i∈ℐ

(5.39)

So, F, F+ are convex functions on ℝJ+ . By definition (5.12) Ξ0ζ (P)⃗ = μ(F(ζ ̄ )) 2 Since w⃗ ∈ ℝJ++ by assumption.

̄ ⃗ (resp. Ξ0,+ ζ (P) = μ(F+ (ζ ))).

(5.40)

5.3 Dominance | 67

Next, using (5.30) we can write M⃗ as m⃗ m⃗ M⃗ = (m1 1 , . . . , mN N ) . m1 mN Then m⃗ P⃗ : M⃗ = ∑ mi p⃗ i ⋅ i . mi i∈ℐ

(5.41)

By definition F(

m⃗ i m⃗ ) ≥ p⃗ i ⋅ i mi mi

resp. F+ (

m⃗ i m⃗ ) ≥ [p⃗ i ⋅ i ] mi mi +

(5.42)

for any i ∈ ℐ . From (5.38), (5.40)–(5.42) we obtain a contradiction to (5.36). Let us extend the definition of dominance from the set of N × J matrices 𝕄+ (N, J) to the set of (ℝJ+ )-valued functions on the general measure space X. Definition 5.3.2. Let μ̄ = (μ(1) , . . . , μ(J) ), ν̄ = (ν(1) , . . . , ν(J) ) be a pair of ℝJ -valued measures on measure spaces X, Y, respectively. We have (X, μ)̄ ≻ (Y, ν)̄ iff there exists a measure π ∈ ℳ+ (X × Y) such that ∫ x∈X

dμ(j) (x)π(dxdy) = ν(j) (dy), dμ

j = 1, . . . , J,

where μ = ∑J1 μ(j) . The following theorem is an extension of Theorem 5.2. Some version of it appears in Blackwell [7]. Theorem 5.3. We have (X, μ)̄ ≻ (Y, ν)̄ iff ∫F ( X

dμ̄ dν̄ ) dμ ≥ ∫ F ( ) dν, dμ dν

(5.43)

X

for any convex F : ℝJ → ℝ. Here ν := ∑Jj=1 ν(j) . Letting F(x)⃗ := ±1⃗ ⋅ x⃗ we obtain from Theorem 5.3 the following. Corollary 5.3.1. A necessary condition for the dominance (X, μ)̄ ≻ (Y, ν)̄ is the balance condition ̄ ) = ν(Y ̄ ). μ(X

By Theorem 5.2 (and its special case in Proposition 5.2) we obtain the following characterization.

68 | 5 Weak multipartitions ̄ Then ΔN (μ)̄ ⊇ ΔN (η)⃗ for any N ∈ ℕ. Corollary 5.3.2. Let (X, μ)̄ ≻ (Y, ν). In fact, the other direction holds as well. Theorem 5.4. We have (X, μ)̄ ≻ (Y, ν)̄ iff ΔN (μ)̄ ⊇ ΔN (η)⃗ for any N ∈ ℕ. Definition 5.3.3. Two weak partitions μ⃗ = (μ1 , . . . , μN ), ν⃗ = (ν1 , . . . , νN ) of (X, μ) and (Y, ν), respectively, are μ̄ − ν̄ congruent iff ∫ X

dμ̄ dν̄ dμ = ∫ dνi , dμ i dν

i = 1, . . . , N.

Y

We denote this relation by μ⃗ ⊗ μ̄ ∼ ν⃗ ⊗ ν.̄ We may now reformulate Theorem 5.4 as follows. Theorem 5.5. We have (X, μ)̄ ≻ (Y, ν)̄ iff for any weak partition ν⃗ of (Y, ν)̄ there exists a partition μ⃗ such that μ,⃗ ν are μ̄ − ν̄ congruent. Proof. The “only if” direction is clear. For the “if” direction, let us consider a sequence of N-partitions νN of ν such that ∫F (

N 1 dν̄ dν̄ ) dν = lim ∑ νiN (Y)F ( N ∫ dνiN ) N→∞ dν νi (Y) dν 1

(5.44)

Y

Y

for any continuous function F. Such a sequence can be obtained, for example, by taking fine strong partitions {ANi } of Y such that νN = μ⌊ANi (Chapter 6). For any such partition let μN be the congruent μ̄ − ν̄ partition μ⃗ of μ. Let now F be a convex function. By Jensen’s inequality ∫F ( X

dμ̄ dμ̄ N 1 ) dμNi ≥ μNi (X)F ( N dμ ) , ∫ dμ μi (X) dμ i X

while μNi (X)F (

1

∫

μNi (X) X

dμ̄ N 1 dν̄ dμ ) = νiN (Y)F ( N ∫ dνiN ) dμ i νi (Y) dν Y

by congruency. Using μ = ∑N1 μNi and summing over i we get inequality (5.43) via (5.44).

5.3.1 Minimal elements Let λ ∈ ℳ+ (Y) and λ(Y) = μ(X). By Theorem 5.3 and the Jensen inequality we obtain the following.

5.3 Dominance | 69

Proposition 5.3. We have ̄ μ(X)λ ≺ μ,̄ where μ̄ is as in (5.1). We can apply this proposition to the discrete spaces X = Y = ℐ := {1, . . . , N}. Let mi := μ({i}), m(j) := μ(j) (X), m⃗ = (m1 , . . . , mN ), and m̄ = (m(1) , . . . , m(J) ). Consider the set (j) ̄ := {M⃗ ∈ 𝕄+ (N, J); ∑ m(j) Π(m,⃗ m) = m(j) ∀j ∈ 𝒥 ; ∑ mi = mi ∀i ∈ ℐ } . i i∈ℐ

j∈𝒥

From Proposition 5.3 we have the following corollary. ̄ with respect to Corollary 5.3.3. λ(j) ({i}) := {mi m(j) /m} is the minimal point in Π(m,⃗ m) (j) the order relation ≻, where m = ∑i∈ℐ mi . That is, for any ν̄ satisfying ν(j) ({i}) = mi , ̄λ ≺ ν.̄

6 Strong multipartitions 6.1 Strong partitions as extreme points A strong N-subpartition A⃗ of X is a subpartition of X into N measurable subsets which are essentially disjoint: A⃗ := (A1 , . . . , AN ),

Ai ∈ 𝒜(X);

⋃ Ai ⊂ X,

μ(Aj ∩ Ai ) = 0

i∈ℐ

for i ≠ j.

The set of all strong subpartitions of X is denoted by 𝒮𝒫

N

:= {A;⃗ A⃗ is a strong N subpartition of X} .

(6.1)

A strong N-partition is a strong N-subpartition of X which satisfies μ(⋃N1 Ai ) = μ(X). We denote the set of all strong N-partitions by 𝒫 N . We shall omit the index N where no confusion is expected. For any 𝒦 ⊂ 𝕄+ (N, J), the set of 𝒦-valued strong subpartitions is { { { {

} } } }

ζ N 𝒮𝒫 𝒦 := {A⃗ ∈ 𝒮𝒫 , (∫ ζ ̄ dμ, . . . , ∫ ζ ̄ dμ) ∈ 𝒦 } A1

AN

(6.2)

and the set of strong 𝒦-valued partitions is 𝒫 ζ 𝒦 . These definitions should be compared with (5.2). Note that 𝒫 ζ 𝒦 can be embedded w,ζ w,ζ in 𝒫𝒦 in a natural way. Just define μ⃗ = (μ1 , . . . , μN ) ∈ 𝒫𝒦 by μi := μ⌊Ai , i. e., the w,ζ ̄

ζ

restriction of μ to Ai . Likewise, 𝒮𝒫 𝒦 is embedded in 𝒮𝒫 𝒦 . Motivated by the above we extend the definition of M⃗ ζ (5.3) to strong (sub)partitions: M⃗ ζ (A)⃗ := {∫ ζj dμ} ∈ 𝕄+ (N, J).

(6.3)

Ai

Now, we are in a position to generalize (5.1.2) to the strong feasibility sets ΔsN (μ)̄ := {M⃗ ∈ 𝕄+ (N, J); 𝒫 ζ M⃗ ≠ 0} ;

ζ ΔsN (μ)̄ := {M⃗ ∈ 𝕄+ (N, J); 𝒮𝒫 ⃗ ≠ 0} . M

(6.4)

By the remark above we immediately observe that ΔsN (μ)̄ ⊆ ΔN (μ)̄

and ΔsN (μ)̄ ⊆ ΔN (μ)̄

(recall (5.1.2)). These inclusions are, in fact, equalities. Theorem 6.1. We have ̄ ΔsN (μ)̄ = ΔN (μ)̄ and ΔsN (μ)̄ = ΔN (μ). https://doi.org/10.1515/9783110635485-006

(6.5)

72 | 6 Strong multipartitions ̄ Thus we omit, from now on, the index s from ΔsN (μ)̄ and ΔsN (μ). ̄ then 𝒫 w,ζ⃗ is not Proof. We have to prove the opposite inclusion of (6.5). If M⃗ ∈ ΔN (μ), w,ζ ⃗ {M}

empty. By the Radon–Nikodym theorem, any μ⃗ = (μ1 , . . . , μN ) ∈ 𝒫

{M}

is characterized

by h⃗ = (h1 , . . . , hN ), where hi are the Radon–Nikodym derivatives of μi with respect to w,ζ μ, namely, μi = hi μ. Since μi ≤ μ, 0 ≤ hi ≤ 1 μ-a. e. on X. Now 𝒫 ⃗ is convex and {M}

compact in the weak* topology C ∗ (X) (Appendix B.3). By the Krein–Milman theorem w,ζ (Appendix A.1) there exists an extreme point μ⃗ in 𝒫 ⃗ . We show that an extreme point {M}

is a strong partition, namely, hi ∈ {0, 1} μ-a. e. on X, for all i ∈ ℐ . Since ∑N hi = 1 μ-a. e. w,ζ on X for any μ⃗ ∈ 𝒫 ⃗ , it is enough to show that for i ≠ j, hi and hj cannot both be {M}

positive on a set of positive μ measure. Let ϵ > 0, let B ⊂ X be measurable, and let μ(B) > 0 such that both hj > ϵ and hi > ϵ for some i ≠ j. Since hj + hi ∈ [0, 1] it follows also that hj , hi are smaller than 1 − ϵ on B as well. The vector-Lyapunov convexity theorem states that the range of a non-atomic vector measures with values in a finite-dimensional space is compact and convex [32]. In particular the set } { R(B) := {(∫ ζ ̄ dμ1 , . . . , ∫ ζ ̄ dμN ) ; A ∈ 𝒜(X), A ⊂ B} ⊂ 𝕄+ (N, J) A } { A is compact and convex. Obviously, R(B) contains the zero point 0⃗ ∈ 𝕄+ (N, J) since ⃗ 0 ⊂ B and μ(0) = 0.⃗ Hence we can find a subset C ⊂ B such that ∫ ζj dμ(j) = C

1 ∫ ζ dμ(j) , 2 j B

∫ ζi dμi =

1 ∫ ζ dμ . 2 i i B

C

Set w := 1B − 2 × 1C , where 1A stands for the characteristic function of a measurable set A ⊂ X. It follows that w is supported on B, |w| ≤ 1, and μ(j) (wζj ) = μi (wζi ) = 0.

(6.6)

By assumption, hj (x) ± ϵw(x) ∈ [0, 1] and hi (x) ± ϵw(x) ∈ [0, 1] for any x ∈ B. Let ν⃗ := (ν1 , . . . , νN ), where νj = μ(j) , νi = −μi , and νk = 0 for k ≠ j, i. Let μ⃗ 1 := μ⃗ + ϵwν,⃗

w,ζ μ⃗ 2 := μ⃗ − ϵwν.⃗ Then by (6.6) both μ⃗ 1 , μ⃗ 2 are in 𝒫 ⃗ and μ⃗ = {M}

contradiction to the assumption that μ⃗ is an extreme point.

1 μ⃗ 2 1

+ 21 μ⃗ 2 . This is in

6.2 Structure of the feasibility sets Assumption 6.2.1. We have μ [x ∈ X; p⃗ ⋅ ζ ̄ (x) = 0] = 0 for any p⃗ ∈ ℝJ − {0}.

(6.7)

6.2 Structure of the feasibility sets | 73

Assumption 6.2.1 is the key to our next discussion on cartels and fixed exchange ratios. Cartels By a coalition we mean a subset of agents ℐ 󸀠 ⊂ ℐ which decide to join together and form a cartel. By a cartel we mean that the price vector −p⃗ I 󸀠 ∈ ℝJ for the list of goods is the same for all members of the coalition ℐ 󸀠 . That is, p⃗ i ≡ p⃗ ℐ 󸀠

∀ i ∈ ℐ󸀠.

The capacity of a coalition ℐ 󸀠 is just the sum of the capacities of its members, m⃗ ℐ 󸀠 = ∑ m⃗ i . i∈ℐ 󸀠

The price vector −p⃗ ℐ 󸀠 for the coalition ℐ 󸀠 is determined by the capacity m⃗ ℐ 󸀠 of this coalition (and these of the other coalitions, if there are any) via the equilibrium mechanism. Exchange ratio A fixed price ratio emerges whenever the agent recognizes a fixed exchange rate between the goods J. Suppose the agent can exchange one unit of the good j for z units of the good k. This implies that the price −pj she charges for j is just −zpk , where −pk is the price she charges for k. More generally, if z⃗ := (z1 , . . . , zJ ) is a fixed vector such that z = zk /zj is the exchange rate of j to k, then the price vector charged by this agent is a multiple p⃗ = qz,⃗ where the reference price q ∈ ℝ is determined, again, by the equilibrium mechanism.

6.2.1 Coalitions and cartels Assume the set of agents ℐ is grouped into a coalition ensemble, that is, a set D = {ℐi } of disjoint coalitions: Given such an ensemble D, no agent can be a member of two different coalitions, that is, ℐn ∩ ℐn󸀠 = 0

for n ≠ n󸀠 ,

and any agent is a member of some coalition ⋃ ℐn = ℐ . n

74 | 6 Strong multipartitions Definition 6.2.1. Given M⃗ := (m⃗ 1 , . . . , m⃗ N ) ∈ 𝕄+ (N, J) and a coalition ensemble D composed of k := |D| coalitions D = (ℐ1 , . . . , ℐk ) ⃗ := (m⃗ ℐ , . . . , m⃗ ℐ ) ∈ 𝕄+ (N, k), D(M) 1 k

m⃗ ℐn := ∑ m⃗ i . i∈ℐn

For such a coalition ensemble, the cartel price vector corresponding to P⃗ ∈ 𝕄 (N, k) is 󸀠

D∗ (P)⃗ := (p⃗ 1 , . . . , p⃗ k ) ∈ 𝕄󸀠 (N, J), where p⃗ i ∈ 𝕄󸀠 (N, |ℐi |) is the constant vector whose components are all equal to the i-component of P.⃗ We also consider a partial order on the set of coalition ensembles: D ⪰ D̃ if for each component ℐl ∈ D there exists a component ℐl̃ ∈ D̃ such that ℐl ⊆ ℐj̃ . In particular ̃ |D| ≥ |D|. Note that the grand coalition D = {ℐ } is the minimal one in this order, while the coalition of individuals D = {{i} ∈ ℐ } is the maximal one. By Definition 6.2.1 we obtain the following duality relation between these mappings. Lemma 6.1. For any M⃗ ∈ 𝕄+ (N, J), any coalition set D, and any P⃗ ∈ 𝕄󸀠 (N, |D|) ⃗ D∗ (P)⃗ : M⃗ = P⃗ : D(M). Consider now a coalition ensemble D = {ℐ1 , . . . , ℐ|D| } and a strong (sub)partition A⃗ D := (Aℐ1 , . . . , Aℐ|D| ). Definition 6.2.2. μ⃗ is embedded in A⃗ D if Supp ( ∑ μi ) ⊆ Aℐk i∈ℐk

for k = 1 . . . |D|.

̄ Then there exists a unique maxiTheorem 6.2. Under Assumption 6.2.1, let M⃗ ∈ 𝜕ΔN (μ).

mal1 coalition ensemble D and a unique strong subpartition 𝒜D such that any μ⃗ ∈ 𝒮𝒫

is embedded in 𝒜D . ⃗ is an extreme point in D(Δ (μ)). Moreover, D(M) N ̄

w,ζ ̄ ⃗ {M}

The full proof of Theorem 6.2 is given in Section 6.2.3.

1 That is, there is no coalition ensemble D̃ ⪰ D and a corresponding strong partition A⃗ D̃ correspond⃗ ing to D(M).

6.2 Structure of the feasibility sets | 75

6.2.2 Fixed exchange ratio Suppose now each agent i ∈ ℐ fixes the ratios of the prices she charges for the list of goods J. For this, she determines a vector z⃗(i) := (z1(i) , . . . , zJ(i) ) ∈ ℝJ . The prices p⃗ i := (p(1) , . . . , p(J) ) she charges her customers are multiples of z⃗(i) : i i p⃗ i = qi z⃗(i) ,

qi ∈ ℝ.

Definition 6.2.3. Given M⃗ := (m⃗ 1 . . . m⃗ N ) ∈ 𝕄+ (N, J), let ⃗ := (z⃗(1) ⋅ m⃗ 1 , . . . , z⃗(N) ⋅ m⃗ N ) ∈ ℝN . Z(⃗ M) The dual operation Z⃗ ∗ : ℝN 󳨃→ 𝕄󸀠 (N, J) acting on q⃗ := (q1 , . . . , qN ) ∈ ℝN is defined by Z⃗ ∗ (q)⃗ := (q1 z⃗(1) , . . . , qN z⃗(N) ) . The duality Z,⃗ Z⃗ ∗ satisfies the following. Lemma 6.2. For any M⃗ ∈ 𝕄+ (N, J) and any q⃗ ∈ ℝN , ⃗ Z⃗ ∗ (q)⃗ : M⃗ = q⃗ ⋅ Z(⃗ M). ⃗ N (μ)) ⃗ ̄ (resp. Z(Δ ̄ are closed convex sets in ℝN . We also By Proposition 5.1, Z(Δ N (μ))) observe that ⃗ ⃗ ̄ ⊆ Z(𝜕Δ ̄ 𝜕Z(Δ N (μ)) N (μ)).

(6.8)

This inclusion is strict, in general. Assumption 6.2.2. z⃗(i) ∈ ℝJ , i = i . . . N are pairwise independent (that is, αz⃗(i) + βz⃗(i ) = 0 for i ≠ i󸀠 iff α = β = 0). In addition, z⃗(i) ⋅ ζ ̄ (x) > 0 for any x ∈ X, i ∈ ℐ . 󸀠

Theorem 6.3. Under Assumptions 6.2.1 and 6.2.2: ⃗ ̄ contained in ℝN++ is composed of extreme points of i) The boundary of Z(Δ N (μ)) ⃗ ̄ Z(Δ N (μ)). ⃗ ̄ there exists a unique subpartition of X associated with this point, ii) If m⃗ ∈ 𝜕Z(Δ N (μ)) and this subpartition is a strong one. In particular there is a unique M⃗ ∈ ΔN (μ)̄ such ⃗ that m⃗ = Z(⃗ M).

Remark 6.2.1. Note that unlike Corollary 5.2.2, ⃗ N (μ)) ⃗ ̄ ⊄ 𝜕Z(Δ ̄ Z(Δ N (μ)) in general.

76 | 6 Strong multipartitions 6.2.3 Proofs Recall the definitions (5.10)–(5.13) of ξζ0 , ξζ0,+ and Ξ0ζ , Ξ0,+ . For any P⃗ = (p⃗ 1 , . . . , p⃗ N ) ∈ ζ

𝕄󸀠 (N, J) consider

Ai (P)⃗ := {x ∈ X; p⃗ i ⋅ ζ ̄ (x) > max p⃗ k ⋅ ζ ̄ (x)} ,

(6.9)

A+0 (P)⃗ := {x ∈ X; max p⃗ i ⋅ ζ ̄ (x) ≤ 0} ,

(6.10)

⃗ A+i (P)⃗ := Ai (P)⃗ − A+0 (P).

(6.11)

k∈ℐ;k =i̸

i∈ℐ

We first need the following result. Lemma 6.3. Under Assumption 6.2.1, if P⃗ := (p⃗ 1 , . . . , p⃗ N ) such that p⃗ i ≠ p⃗ i󸀠 for i ≠ i󸀠 , then Ξ0ζ (resp. Ξ0,+ ) is differentiable at P⃗ and ζ ∇p⃗ i Ξ0ζ = ∫ ζ ̄ dμ,

resp. ∇p⃗ i Ξ0,+ = ∫ ζ ̄ dμ. ζ

(6.12)

A+i (P)⃗

Ai (P)⃗

⃗ these sets are mutually essentially disjoint. By AsProof. By the definition of {Ai (P)}, ⃗ = μ(X). Moreover, sumption 6.2.1 and the assumption on P⃗ we obtain that μ(⋃N1 Ai (P)) ∇p⃗ i ξζ = {

ζ ̄ (x) 0

⃗ if x ∈ Ai (P), ⃗ if ∃j ≠ i, x ∈ Aj (p).

(6.13)

In particular, the p⃗ i derivatives of ξζ exist μ-a. e. in X, ∇p⃗ ξζ ∈ 𝕃1 (X; 𝕄+ (N, J)) for any P⃗ ∈ 𝕄󸀠 (N, J) and the partial derivatives are uniformly integrable. Since Ξ0ζ := μ(ξζ ) by definition, its derivatives exist everywhere and ⃗ = ∫ ζ ̄ dμ. ∇p⃗ i Ξ0ζ = μ (∇p⃗ i ξζ (⋅, P)) Ai (P)⃗

Finally, note that Ξ0ζ is a convex function, and the existence of its partial derivatives implies its differentiability (A.10). ⃗ i ∈ ℐ , while In the case of Ξ0,+ we observe that (6.13) still holds for ξζ+ and A+i (P), ζ + + ⃗ + ∇p⃗ ξ = 0 for any x ∈ A (P). Since μ(⋃ A ) = μ(X) we obtain the same result for i

ζ

0

i∈ℐ∪{0}

⃗ i ∈ ℐ. the subpartition induced by {A+i (P)},

i

⃗ is differentiable Corollary 6.2.1. Under Assumption 6.2.2, the function q⃗ → Ξ0,+ (Z⃗ ∗ (q)) ζ

at any q⃗ ∈ ℝN++ .

Proof of Theorem 6.2. Given a price matrix P⃗ := (p⃗ 1 , . . . , p⃗ N ), we associate with p⃗ the ⃗ The collection of pairwise disjoint coalitions defined coalitions ℐp⃗ := {k ∈ ℐ ; p⃗ i = p}. in this way constitutes the ensemble of coalitions DP⃗ : DP⃗ := (ℐ1 , . . . , ℐ|D(P)| ⃗ ),

(6.14)

6.2 Structure of the feasibility sets | 77

where each ℐi coincides with one of the (non-empty) coalitions ℐp⃗ , p⃗ ∈ {p⃗ 1 , . . . , p⃗ N }. ̄ We now recall Theorem 5.1 and Corollary 5.2.1: Iff M⃗ is a boundary point of ΔN (μ), 󸀠 󸀠 ⃗ ⃗ then there exists a non-zero P0 ∈ 𝕄 (N, J) such that, for any P ∈ 𝕄 (N, J), 0,+ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ Ξ0,+ ζ (P0 ) − P0 : M = 0 ≤ Ξζ (P) − P : M.

(6.15)

For any such (possibly non-unique) P⃗ 0 we associate the coalition set D := DP⃗ as 0 defined in (6.14). If there is another P⃗ 󸀠 ≠ P⃗ 0 maximizing (6.15), then by convexity of Ξ0 , (1 − ϵ)P⃗ 0 + ϵP⃗ 󸀠 is a maximizer as well for any ϵ ∈ [0, 1]. By Definition 6.2.1 we get ζ

that for ϵ > 0 sufficiently small DϵP⃗ 󸀠 +(1−ϵ)P⃗ ≻ D, and the pair of coalition ensembles 0 agrees iff D = DP⃗ 󸀠 . Thus, the maximal coalition ensemble is unique. Let (μ1 , . . . , μ|D| ) be a subpartition associated with the maximal coalition D. In particular, ⃗ D(M) = μi (ζj ), i (j)

i ∈ D, j ∈ 𝒥 .

By the definition of Ξ0,+ (5.13), (6.15), and Lemma 6.1 we get ζ 0,+ ∗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ Ξ0,+ ζ (P0 ) − P0 : M = Ξζ (D (P0 )) − P0 ⋅ D(M)

≡ ∑ μi [ξζ0,+ (x, P⃗ 0 ) − P⃗ 0,i ⋅ ζ ̄ ] + μ0 [ξζ0,+ (⋅, P⃗ 0 )] = 0, i∈D

(6.16)

where μ0 := μ−∑i∈D μi . From the definition of ξζ0,+ we obtain that ξζ0,+ (x, P⃗ 0 ) ≥ p⃗ i,0 ⋅ ζ ̄ (x) or any (x, i) ∈ X × D as well as ξ 0,+ ≥ 0 on X. Thus, (6.16) implies that ξ 0,+ (x, P⃗ 0 ) = ζ

p⃗ i,0 ⋅ ζ ̄ (x) a. e. (μi ), as well as ξζ0,+ (x, P⃗ 0 ) = 0 a. e. (μ0 ).

ζ

On the other hand, we get via (6.11), (6.10) adapted to D that ξζ0,+ (x, P⃗ 0 ) > p⃗ i,0 ⋅ ζ ̄ (x) if x ∈ A+k (P⃗ 0 ), where for any k ∈ D ∪ {0} − {i}. Hence Supp(μi ) ⊂ X −

⋃ k∈D∪{0};k =i̸

A+k (P⃗ 0 ).

(6.17)

Since, by definition, the components of P⃗ 0 are pairwise different we get by Assumption 6.2.1 and (6.11), (6.10) that ⋃i∈D∪{0} A+i (P⃗ 0 ) = X. This and (6.17) imply that μi is the restriction of μ to A+i (P⃗ 0 ), and hence it is a strong partition. The uniqueness of this partition follows as well. Finally, it follows from (6.15) that M⃗ ∈ 𝜕P⃗ Ξ0,+ . Since P⃗ 0 ≠ 0 it follows from 0 ζ 0,+ ⃗ ⃗ Lemma 6.3 that Ξ is differentiable at P0 . Hence M is an extreme point via Proposition A.13.

ζ

The following corollary to the proof of Theorem 6.2 refers to the case of maximal coalition (Definition 6.2.1).

78 | 6 Strong multipartitions Corollary 6.2.2. If P⃗ 0 := (p⃗ 1 , . . . , p⃗ N ) satisfies p⃗ i ≠ p⃗ j for any i ≠ j, then there exists a unique partition in 𝒫 ζ {M⃗ } , where M⃗ 0 = ∇P⃗ Ξ0ζ . Moreover, this partition is a strong one, 0 0 given by (6.9), where P⃗ 0 is substituted for P.⃗ ⃗ ̄ ∩ ℝN++ . Proof of Theorem 6.3. i) Assume that m⃗ ∈ 𝜕Z(Δ N (μ)) 󸀠 −1 ⃗ ∩ 𝜕ΔN (μ). ̄ By Theorem 5.1 we get Let M⃗ ∈ Z⃗ (m) ⃗ ⃗ ⃗󸀠 Ξ0,+ ζ (P) − P : M ≥ 0 for any P⃗ ∈ 𝕄󸀠 (N, J). In particular, we substitute P⃗ = Z⃗ ∗ (q)⃗ and we get, for any q⃗ ∈ ℝN , ⃗∗ ⃗ ⃗∗ ⃗ ⃗󸀠 Ξ0,+ ζ (Z (q)) − Z (q) : M ≥ 0. From Lemma 6.2 (and since m⃗ = Z(⃗ M⃗ 󸀠 ) by definition), 0,+ ⃗ ∗ ⃗∗ ⃗ ⃗ ⃗ ⃗󸀠 ⃗ ⃗ ⃗ Ξ0,+ ζ (Z (q)) − q ⋅ Z(M ) = Ξζ (Z (q)) − q ⋅ m ≥ 0

(6.18)

⃗ ̄ we get, as in the proof of Corolfor any q⃗ ∈ ℝN . Since, in addition, m⃗ ∈ 𝜕Z(Δ N (μ)), lary 5.2.1, that there exists a non-zero q⃗ 0 ∈ ℝN for which ⃗∗ ⃗ ⃗ ⃗ Ξ0,+ ζ (Z (q0 )) − q0 ⋅ m = 0.

(6.19)

We prove now that q⃗ 0 ∈ ℝN++ . Surely, it is impossible that all components of q⃗ 0 are non-positive. Assume with no limitation of generality that q0,1 > 0. By Assumption 6.2.1 we can find δ > 0 such ⃗ x) ≥ q0,1 z⃗(1) ⋅ ζ ̄ (x) > δq0,1 on X. that z⃗(1) ⋅ ζ ̄ (x) > δ for any x ∈ X. Then ξζ (Z⃗ ∗ (q), Suppose q0,j ≤ 0 for some j ≠ 1, and let ϵ > 0 for which ϵz⃗(j) ⋅ ζ ̄ < δq0,1 on X. Then 0 ⃗∗ ξζ (Z (q⃗ 0 ), x) = ξζ0 (Z⃗ ∗ (q⃗ 0 + ϵej⃗ ), x) on X. Here e⃗j is the unit coordinate vector pointing in the positive j-direction. Indeed, both q0,j z⃗(j) ⋅ ζ ̄ (x) and (q0,j + ϵ)z⃗(j) ⋅ ζ ̄ (x) are smaller than q⃗ 0,1 z⃗(1) ⋅ ζ ̄ (x) for any x ∈ X, so the j-component does not contribute to the value of ξζ0 at any point x ∈ X. Hence Ξ0ζ (q⃗ 0 ) = Ξ0ζ (Z⃗ ∗ (q⃗ 0 + ϵe⃗j )). So Ξ0ζ (Z⃗ ∗ (q⃗ 0 + ϵe⃗j )) − (q⃗ 0 + ϵe⃗j ) ⋅ m⃗ = Ξ0ζ (Z⃗ ∗ (q⃗ 0 )) − q⃗ 0 ⋅ m⃗ − ϵmj = −ϵmj < 0 by (6.19) (recall mj > 0 by assumption). This contradicts (6.18), and hence q0,j > 0 as well as q⃗ ∈ ℝN+ . ⃗ ⃗ ̄ ∩ ℝN++ is an extreme point in Z(Δ ̄ Consider We now prove that m⃗ ∈ 𝜕Z(Δ N (μ)) N (μ)). 0,+ ⃗ ∗ ⃗ By Assumptions 6.2.1 and 6.2.2 and Corollary 6.2.1 we the function q⃗ → Ξζ (Z (q)).

observe that this function is convex and differentiable at any q⃗ ∈ ℝN++ . Its essential ⃗ ̄ Thus, (6.19) and Proposition A.13 imply that m⃗ is an extreme point domain is Z(Δ N (μ)). ⃗ ̄ of Z(ΔN (μ)). ii) Let now (μ1 , . . . , μN ) be a partition associated with m.⃗ In particular, ∫ z⃗(i) ⋅ ζ ̄ dμi = (5.13) we get mi . By the definition of Ξ0,+ ζ 0,+ ⃗∗ ⃗ ⃗∗ ⃗ ⃗ ⃗ ⃗(i) ̄ Ξ0,+ ζ (Z (q0 )) − q0 ⋅ m ≡ μ [ξζ (x, Z (q0 ))] − ∑ q0,i μi [z ⋅ ζ ] = 0. i∈ℐ

6.2 Structure of the feasibility sets | 79

On the other hand, since μ ≥ ∑i∈ℐ μi , we get N

μ0 (ξζ0,+ (x, Z⃗ ∗ (q⃗ 0 ))) + ∑ μi (ξζ0,+ (x, Z⃗ ∗ (q⃗ 0 )) − qi,0 z⃗(i) ⋅ ζ ̄ ) = 0, i=1

0,+

where μ0 = μ − ∑i∈ℐ μi . Since ξ ≥ 0 by definition (5.11), we obtain, in particular, that ξ +,0 (x, Z⃗ ∗ (q⃗ 0 )) = 0 μ0 a. e. Thus, μ0 is supported in A+0 (Z⃗ ∗ (q⃗ 0 )) via (6.10). From the definition (5.11) of ξζ0,+ we also obtain that ξζ0,+ (x, Z⃗ ∗ (q⃗ 0 )) ≥ qi,0 z⃗(i) ⋅ ζ ̄ (x) for any x ∈ X. Thus, ξ 0,+ (x, Z⃗ ∗ (q⃗ )) = q z⃗(i) ⋅ ζ ̄ (x) for μ -a. e. x. 0

ζ

i,0

i

On the other hand, from (6.11) (substitute Z⃗ ∗ (q)⃗ for P)⃗ we get ξζ0,+ (x, Z⃗ ∗ (q⃗ 0 )) < ⃗ for any k ≠ i. Hence qk,0 z⃗(k) ⋅ ζ ̄ (x) μi a. s. if x ∈ A+ (Z⃗ ∗ (q)) k

Supp(μi ) ⊂ X − ⋃ A+k (Z⃗ ∗ (q⃗ 0 )). k =i̸

(6.20)

By (6.9)–(6.11) we obtain that the union of A+i (Z⃗ ∗ (q⃗ 0 )), i ∈ {0} ∪ ℐ , is of full μ-measure. This and (6.20) imply that μi is the restriction of μ − μ0 to Ai (Z⃗ ∗ (q⃗ 0 )), and hence it is a strong subpartition. The uniqueness follows since the same reasoning holds for any subpartition corresponding to m.⃗ ⃗ N (μ)) ̄ the set Z(Δ ̄ ⊂ ℝN+ may contain interior points (compare with CorolNote that unlike ΔN (μ), lary 5.2.2).

Proposition 6.1. Under Assumptions 6.2.1 and 6.2.2, ⃗ ⃗ N (μ)). ̄ ⊂ 𝜕Z(Δ ̄ ℝN++ ∩ 𝜕Z(Δ N (μ)) ⃗ ̄ ∩ ℝN++ is a strong partiIn particular, any subpartition corresponding to m⃗ ∈ 𝜕Z(Δ N (μ)) tion. ⃗ ̄ ∩ ℝN++ . Following the proof of Theorem 6.3 we get the exisProof. Let m⃗ ∈ 𝜕Z(Δ N (μ)) w,ζ ̄ tence of q⃗ 0 ∈ ℝN satisfying (6.19). If μ⃗ ∈ Z(⃗ 𝒮𝒫 ), then ⃗ {m}

++

0,+ 0,+ ⃗∗ ⃗ ⃗∗ ⃗ ⃗∗ ⃗ Ξ0,+ ζ (Z (q0 )) ≡ μ [ξζ (x, Z (q0 ))] ≥ ∑ μi [ξζ (⋅, Z (q0 ))] i∈ℐ

≥ ∑ q0,i ⋅ μi (z⃗ i∈ℐ

(i)

⃗∗ ⃗ ⃗ ≡ Ξ0,+ ⋅ ζ ̄ ) = q⃗ 0 ⋅ Z(⃗ m) ζ (Z (q0 )).

In particular, μ (ξζ0,+ (x, Z⃗ ∗ (q⃗ 0 ))) = ∑ μi (ξζ0,+ (x, Z⃗ ∗ (q⃗ 0 ))) . i∈ℐ

Since q⃗ 0 ∈ ℝN++ , Assumption 6.2.1 and the definition of ξζ0,+ imply that ξζ0,+ (⋅, Z⃗ ∗ (q⃗ 0 )) is positive and continuous on X. This and ∑i∈ℐ μi ≤ μ imply that, in fact, ∑i∈ℐ μi = μ, so w,ζ ⃗ N (μ)). ⃗ N (μ)) ⃗ ̄ Since Z(Δ ̄ ∩ 𝜕Z(Δ ̄ ⊂ μ⃗ ∈ 𝒫{m}⃗ is a strong partition. In particular, m⃗ ∈ Z(Δ N (μ)) ⃗ N (μ)), ⃗ N (μ)) ̄ m⃗ ∈ 𝜕Z(Δ ̄ as well. 𝜕Z(Δ

80 | 6 Strong multipartitions 6.2.4 An application: two states for two nations Suppose X is a territory held by two ethnic groups living unhappily together, say, 𝒥 and 𝒫 . Let μ be the distribution of the total population in X. Let ζ𝒥 : X → [0, 1] be the relative density of population 𝒥 . Then ζ𝒫 := 1 − ζ𝒥 is the relative density of the population 𝒫 . It was suggested by some wise men and women that the territory X should be divided between the two groups, to establish a 𝒥 -state A𝒥 and a 𝒫 -state A𝒫 : A𝒥 ⊂ X, A𝒫 ⊂ X;

A𝒥 ∪ A𝒫 = X,

μ(A𝒥 ∩ A𝒫 ) = 0.

Under the assumption that nobody is forced to migrate from one point to another in X, what are the possibilities of such divisions? The question can be reformulated as follows. Let us assume that an A𝒥 state is formed whose 𝒥 population is m𝒥 and whose 𝒫 population is m𝒫 :2 ∫ ζ𝒥 dμ = m𝒥 , A𝒥

∫ ζ𝒫 dμ = m𝒫 . A𝒥

The evident constraints are 0 ≤ m𝒥 ≤ μ(ζ𝒥 );

0 ≤ m𝒫 ≤ μ(ζ𝒫 ).

(6.21)

Assuming for convenience that the total population μ is normalized (μ(X) = 1, so μ(ζ𝒥 ) + μ(ζ𝒫 ) = 1), we may use Theorem 5.2 to characterize the feasibility set S in the rectangle domain (6.21) by (m𝒥 , m𝒫 ) ∈ S ⇔ μ (F(ζ𝒥 , ζ𝒫 )) ≥ (m𝒥 + m𝒫 )F (

μ(ζ𝒥 ) − m𝒥 , μ(ζ𝒫 ) − m𝒫 m𝒥 , m𝒫 ) + (1 − m𝒫 − m𝒥 )F ( ). m𝒥 + m𝒫 1 − m𝒥 − m𝒫

(6.22)

From Proposition 5.3 we also obtain that the diagonal of the rectangle (6.21) is always contained in S: ⋃ α(μ(ζ𝒥 ), μ(ζ𝒫 )) ⊂ S.

α∈[0,1]

See Figure 6.1. What else can be said about the feasibility set S, except that it is convex and contains the diagonal of the rectangle (6.21)? If μ(ζ1 (x)/ζ2 (x) = r) = 0 for any r ∈ [0, ∞], then the assumption of Theorem 6.3 is satisfied with J = 2, z⃗(1) = (1, 0), z⃗(2) = (0, 1). In particular we obtain the following. 2 Of course, the 𝒥 − 𝒫 populations of the 𝒫 state are, respectively, M𝒥 − m𝒥 and M𝒫 − m𝒫 .

6.3 Further comments |

81

Figure 6.1: Projection on the diagonal.

Proposition 6.2. All points of the boundary 𝜕S ∩ ℝ2++ are extreme points. For each (m𝒥 , m𝒫 ) ∈ 𝜕S ∩ ℝ2++ there exists r ∈ [0, ∞] such that the corresponding partition A𝒥 := {x ∈ X; ζ𝒥 (x)/ζ𝒫 (x) ≥ r} ,

A𝒫 := {x ∈ X; ζ𝒥 (x)/ζ𝒫 (x) ≤ r}

is unique. In particular, S is contained in the parallelogram inf

x∈X

ζ𝒥 (x) m𝒥 μ(ζ𝒥 ) − m𝒥 m𝒥 μ(ζ𝒥 ) − m𝒥 ζ𝒥 (x) ≤ ∧ ≤ ∨ ≤ sup . ζ𝒫 (x) m𝒫 μ(ζ𝒫 ) − m𝒫 m𝒫 μ(ζ𝒫 ) − m𝒫 x∈X ζ𝒫 (x)

6.3 Further comments ⃗ The special case of Theorem 6.2 where N = 1 can be formulated as follows: Let σ(dx) := ̄ζ (x)μ(dx) be an ℝJ -valued measure on X. The set Δ (μ)̄ corresponds, in that case, to N the image of σ⃗ over all measurable subsets of X: ⃗ ΔN (μ)̄ := {σ(A); A ⊂ X is μ measurable } ⊂ ℝJ . The geometry of such sets was discussed by several authors. In particular, several equivalent sufficient and necessary conditions for the strict convexity of ΔN (μ)̄ were introduced in [43, 4, 5]. One of these conditions is the following. Theorem 6.4 ([43, 4]). The set ΔN (μ)̄ is strictly convex iff the following condition holds: ⃗ For any measurable set A ⊂ X for which σ(A) ≠ 0 there exist measurable sets A1 , . . . , AJ ⊂ ⃗ 1 ), . . . , σ(A ⃗ J ) are linearly independent. A such that the vectors σ(A Theorem 6.4 can be obtained as a special case of Theorem 6.2. Indeed, if N = 1, then there is only one possible “coalition,” composed of the single agent, hence the strict convexity of this set (namely, the property that any boundary point is an extreme point) is conditioned on Assumption 6.2.1. Let us show that for a continuous ζ ̄ (5.1), Assumption 6.2.1 is, indeed, equivalent to the assumption of Theorem 6.4.

82 | 6 Strong multipartitions If Assumption 6.2.1 fails, then there exists a non-zero p⃗ ∈ ℝJ and a measurable set A such that μ(A) > 0 and p⃗ ⋅ ζ ̄ (x) = 0 on A. Hence, for any measurable B ⊂ A, ⃗ p⃗ ⋅ σ(B) ≡ ∫B p⃗ ⋅ ζ ̄ dμ = 0 as well. Hence p⃗ is not spanned by any collection of J subsets in A. ⃗ 1) . . . Conversely, suppose μ(A) > 0 and let k be the maximal dimension of Sp(σ(A ⃗ J )), where A1 , . . . , AJ run over all μ-measurable subsets of A. We can find k subsets σ(A ⃗ 1 ) . . . σ(A ⃗ k )) equals k. If k < J, then there A1 . . . Ak of A such that the dimension of Sp(σ(A J ⃗ ⃗ ⃗ exists p ∈ ℝ such that p ⋅ σ(Ai ) = 0 for i ∈ {1, . . . , k}. If there exists a measurable B ⊂ A ⃗ ⃗ i ), i = 1, . . . , k, such that p⃗ ⋅ σ(B) ≠ 0, then the dimension of the space spanned by σ(A ⃗ ⃗ and σ(B) is k + 1. This contradicts the assumed maximality of k. Thus, p⃗ ⋅ σ(B) = 0 for any measurable subset of A, which implies that p⃗ ⋅ ζ ̄ = 0 on A. As a special case of Theorem 6.3 we may consider z⃗(i) to be the principle coordi⃗ ̄ in ℝN is, then, given by nates of ℝN (in particular, J = N). The set Z(Δ N (μ)) } { } { ⃗ ̄ , (∫ ζ dμ, . . . , ζ dμ) Z(Δ ( μ)) = ∫ 1 N N } { } { A A N } { 1 where A⃗ := (A1 , . . . , AN ) runs over the set 𝒮𝒫 N of all strong subpartitions of X (Section 6.1). Such sets are the object of study in [50]. The case of N = 2 is the case we considered in Section 6.2.4. A detailed study of this case can be found in [31].

7 Optimal multipartitions 7.1 Optimality within the weak partitions 7.1.1 Extension to hyperplane w,ζ {M⃗ 0 }

In order to consider the optimization of θ on 𝒫

where M⃗ 0 ∈ ΔN (μ)̄ (resp. on 𝒮𝒫

w,ζ ̄ {M⃗ 0 }

̄ we introduce the following extension of Theorem 5.1. where M⃗ 0 ∈ ΔN (μ)), Let ℚ be a subspace of 𝕄󸀠 (N, J). Let ℚ⊥ ⊂ 𝕄+ (N, J) be the subspace of annihilators of ℚ, that is, ℚ⊥ := {M⃗ ∈ 𝕄+ (N, J), P⃗ : M⃗ = 0 ∀P⃗ ∈ ℚ}. Given such ℚ and P⃗ 0 ∈ 𝕄󸀠 (N, J), the following theorem extends Theorem 5.1 to the hyperplane ℚ + sP⃗ 0 . Theorem 7.1. For any s ∈ ℝ inf Ξ0ζ (Q⃗ + sP⃗ 0 ) − Q⃗ : M⃗ 0 =

⃗ Q∈ℚ

sup

sP⃗ 0 : M⃗

(7.1)

sup

sP⃗ 0 : M.⃗

(7.2)

⊥ +M⃗ )∩Δ (μ) ⃗ M∈(ℚ 0 N ̄

as well as ⃗ ⃗ ⃗ ⃗ inf Ξ0,+ ζ (Q + sP0 ) − Q : M0 =

⃗ Q∈ℚ

⊥ +M⃗ )∩Δ (μ) ⃗ ̄ M∈(ℚ 0 N

The case ℚ = 𝕄󸀠 (N, J) reduces to Theorem 5.1. Indeed, if M⃗ 0 ∈ ̸ ΔN (μ)̄ (resp. M⃗ 0 ∈ ̸ ̄ then the right side of (7.1), (7.2) is a supremum over a null set (since ℚ⊥ = {0}) ΔN (μ)), and, by the definition of the supremum, it equals −∞. If, on the other hand, (ℚ⊥ + M⃗ 0 ) ∩ ΔN (μ)̄ ≠ 0 (resp. (ℚ⊥ + M⃗ 0 ) ∩ ΔN (μ)̄ ≠ 0), then the supremum on the right side of (7.1) (resp. (7.2)) is always attended, since both ̄ ΔN (μ)̄ are compact sets. Thus, there exists M⃗ ∗ ∈ (ℚ⊥ + M⃗ 0 ) ∩ ΔN (μ)̄ (resp. M⃗ ∗ ∈ ΔN (μ), ⊥ ̄ such that (ℚ + M⃗ 0 ) ∩ ΔN (μ)) sP⃗ 0 : M⃗ ∗ =

sup

⊥ +M⃗ )∩Δ (μ) ⃗ M∈(ℚ 0 N ̄

sP⃗ 0 : M⃗ resp. sP⃗ 0 : M⃗ ∗ =

sup

⊥ +M⃗ )∩Δ (μ) ⃗ ̄ M∈(ℚ 0 N

sP⃗ 0 : M.⃗

(7.3)

Remark 7.1.1. We can make a natural connection between reduction to coalition ensembles introduced in Section 6.2.1 and the duality with respect to affine subsets. Indeed, given a coalition D we may define ℚD := D∗ (𝕄󸀠 (N, J)). If we imply Theorem 7.1 in the special case s = 0 (and for arbitrary P⃗ 0 ) we can get Theorem 6.2 from the following statement: For any M⃗ 0 ∈ 𝜕ΔN (μ)̄ there exists a unique maximal coalition ensemble D such that the inequality ⃗ ⃗ ⃗ Ξ0,+ ζ (Q) − Q : M0 ≥ 0 https://doi.org/10.1515/9783110635485-007

84 | 7 Optimal multipartitions holds for any P⃗ ∈ ℚD , and there exists a unique P⃗ ≠ 0 in ℚD along which the above ⃗ α ≥ 0). This P⃗ inequality turns into an equality on the ray spanned by P⃗ (P⃗ = {αP}, ⃗ induces the unique strong subpartition AD . There is also a natural connection between Theorem 6.3 and Theorem 7.1, which is explained below. Let ℚ = Z⃗ ∗ (ℝN ) ⊂ 𝕄󸀠 (N, J). We may imply Theorem 7.1 for s = 0 and get The⃗ ̄ ∩ ℝN+ there exists a orem 6.3 from the following statement: For any m⃗ 0 ∈ 𝜕Z(Δ N (μ)) −1 ⃗ ⃗ unique M0 ∈ Z (m⃗ 0 ) such that the inequality ⃗ ⃗ ⃗ Ξ0,+ ζ (Q) − Q : M0 ≥ 0 holds for any Q⃗ ∈ ℚ, and there exists a unique P⃗ ≠ 0 in ℚ along which the above ⃗ α ≥ 0). This P⃗ inequality turns into an equality on the ray spanned by P⃗ (P⃗ = {αP}, induces the unique strong subpartition. The minimizer Q⃗ ∈ ℚ on the left side of (7.1), (7.2), however, is not necessarily attained. Recall also Definition 4.6.2 of the weak (sub)partition sets 𝒫 w , 𝒮𝒫 w and ̄ (7.3) implies that, for such pairs (M⃗ 0 , P⃗ 0 ) ∈ (5.2). Since M⃗ ∗ ∈ ΔN (μ)̄ (resp. in ΔN (μ)), 󸀠 ⃗ 𝕄+ (N, J) × 𝕄 (N, J), there exist M∗ ∈ ℚ⊥ + M⃗ 0 and (sub)partitions μ⃗ which maximize ⃗ on 𝒫 w⊥ ⃗ (resp. on 𝒮𝒫 w ⊥ ⃗ ), i. e., sP⃗ 0 : M⃗ ζ (μ)) ℚ +M ℚ +M 0

0

sP⃗ 0 : M⃗ ζ (μ)⃗ = inf Ξ0ζ (Q⃗ + sP⃗ 0 ) − Q⃗ : M⃗ 0 , ⃗ Q∈ℚ

resp. ⃗ ⃗ ⃗ ⃗ sP⃗ 0 : M⃗ ζ (μ)⃗ = inf Ξ0,+ ζ (Q + sP0 ) − Q : M0 . ⃗ Q∈ℚ

Letting s = 1 we obtain the following. ̄ Proposition 7.1. For each (P⃗ 0 , M⃗ 0 ) ∈ 𝕄󸀠 (N, J)×𝕄+ (N, J) there exist M⃗ ∗ ∈ ΔN (μ)∩( M⃗ 0 + ⊥ ⊥ w w ̄ M⃗ 0 +ℚ )) and μ⃗ ∈ 𝒫 (resp. μ⃗ ∈ 𝒮𝒫 ) such that μ⃗ maximizes ℚ ) (resp. M⃗ ∗ ∈ Δ (μ)∩( N

M⃗ ∗

w P⃗ 0 : M⃗ ζ (μ)⃗ on 𝒫ℚ (resp. on 𝒮𝒫 w ), and, moreover, ⊥ +M⃗ ℚ⊥ +M⃗ 0

M⃗ ∗

0

P⃗ 0 : M⃗ ζ (μ)⃗ = inf Ξ0ζ (Q⃗ + P⃗ 0 ) − Q⃗ : M⃗ 0 , ⃗ Q∈ℚ

resp. ⃗ ⃗ ⃗ ⃗ P⃗ 0 : M⃗ ζ (μ)⃗ = inf Ξ0,+ ζ (Q + P0 ) − Q : M0 . ⃗ Q∈ℚ

Ξ0ζ

Let us consider (7.1) in the case s = 0 and ℚ ⊂ 𝕄󸀠 (N, J) a subspace ≠ {0}. We view as defined on the subspace ℚ whose dual is the quotient space 𝕄+ (N, J)/ℚ⊥ . The

7.1 Optimality within the weak partitions | 85

⃗ where Q⃗ ∈ ℚ and M⃗ is any action of ℚ on 𝕄+ (N, J)/ℚ⊥ is defined, naturally, as Q⃗ : M, ⊥ representative form 𝕄+ (N, J)/ℚ . Hence (7.1) reads inf Ξ0ζ (Q)⃗ − Q⃗ : M⃗ 0 = 0

⊥ ̄ iff M⃗ 0 ∈ ΔN (μ)/ℚ .

⃗ Q∈ℚ

(7.4)

In the general case we may view (Q,⃗ s) → Ξ0ζ (Q⃗ + sP⃗ 0 ) as a positively homogeneous function on the space ℚ⊗ℝ. The dual of this space is 𝕄+ (N, J)/ℚ⊥ ⊕ℝ, and the duality action is (Q,⃗ t) : (M,⃗ s) := Q⃗ : M⃗ + ts, where Q⃗ ∈ ℚ, M⃗ is any representative from 𝕄+ (N, J)/ℚ⊥ and ts is just the product of t and s in ℝ. Then (7.1) applied to all s ∈ ℝ reads as inf

⃗ (Q,s)∈(ℚ⊕ℝ)

Ξ0ζ (Q⃗ + sP⃗ 0 ) − (Q,⃗ s) : (M⃗ 0 , t) = 0

? ̄ ⊂ 𝕄+ (N, J)/ℚ⊥ ⊗ ℝ, where iff (M⃗ 0 , t) ∈ Δ N (μ) ⊥ ? ̄ := {(M⃗ 0 , t), M⃗ 0 ∈ ΔN (μ)/ℚ ̄ Δ , N (μ)

inf

⊥ +M⃗ ) ⃗ M∈(ℚ 0

cP0 : M⃗ ≤ t ≤

sup

⊥ +M⃗ ) ⃗ M∈(ℚ 0

⃗ . (7.5) P⃗ 0 : M}

Similarly inf

⃗ (Q,s)∈(ℚ⊕ℝ)

⃗ ⃗ ⃗ ⃗ Ξ0,+ ζ (Q + sP0 ) − (Q, s) : (M0 , t) = 0

? ̄ ⊂ 𝕄+ (N, J)/ℚ⊥ ⊗ ℝ, where iff (M⃗ 0 , t) ∈ Δ N (μ) ⊥ ? ̄ := {(M⃗ 0 , t), M⃗ 0 ∈ ΔN (μ)/ℚ ̄ Δ , N (μ)

inf

⊥ +M⃗ ) ⃗ M∈(ℚ 0

P⃗ 0 : M⃗ ≤ t ≤

sup

⊥ +M⃗ ) ⃗ M∈(ℚ 0

⃗ . (7.6) P⃗ 0 : M}

? ? ̄ is the essential domain ̄ (resp. Δ Recalling Proposition A.12 we observe that Δ N (μ) N (μ)) 0 ⃗ ⃗ ⃗ of the Legendre transform of (Q, s) → Ξζ (Q + sP0 ) (resp. (Q,⃗ s) → Ξ0,+ (Q⃗ + sP⃗ 0 )) as ζ ̄ from 𝕄+ (N, J) to functions on ℚ ⊗ ℝ. It is, in fact, an extension of ΔN (μ)̄ (resp. ΔN (μ)) ⊥ 𝕄+ (N, J)/ℚ ⊗ ℝ. Proof of Theorem 7.1 The inequalities inf Ξ0ζ (Q⃗ + P⃗ 0 ) − Q⃗ : M⃗ 0 ≥

⃗ Q∈ℚ

sup

⊥ +M⃗ )∩Δ (μ) ⃗ M∈(ℚ 0 N ̄

P⃗ 0 : M,⃗

⃗ ⃗ ⃗ ⃗ resp. inf Ξ0,+ ζ (Q + P0 ) − Q : M0 ≥ ⃗ Q∈ℚ

sup

⊥ +M⃗ )∩Δ (μ) ⃗ ̄ M∈(ℚ 0 N

P⃗ 0 : M⃗

(7.7)

hold by Theorem 5.1. In order to prove the reverse inequality we need the Hahn– Banach theorem.

86 | 7 Optimal multipartitions Theorem 7.2 (Hahn–Banach). Let V be a real vector space, p : V → ℝ a sublinear function, and f : U → ℝ a linear functional on a linear subspace U ⊆ V s. t. f (x) ≤ p(x) for every x ∈ U. Then there exists a linear functional F ∈ V ∗ s. t. F(u) = f (u)∀u ∈ U and F(x) ≤ p(x)∀x ∈ V. The Hahn–Banach theorem is valid for any linear space. Here we use it for the finite-dimensional space V ≡ 𝕄󸀠 (N, J). Let p(P)⃗ := inf Ξ(P⃗ + Q)⃗ − (P⃗ + Q)⃗ : M⃗ 0 , ⃗ Q∈ℚ

. Note that where Ξ stands for either Ξ0ζ or Ξ0,+ ζ p≥0

on 𝕄󸀠 (N, J)

(7.8)

̄ Recall that a function by Theorem 5.1 since M⃗ 0 ∈ S, where S = ΔN (μ)̄ (resp. S = ΔN (μ)). p is sublinear iff 1. p(sP)⃗ = sp(P)⃗ for any P⃗ ∈ 𝕄󸀠 (N, J) and s > 0, 2. p(P⃗ 1 + P⃗ 2 ) ≤ p(P⃗ 1 ) + p(P⃗ 2 ). Note that Ξ is sublinear by definition (5.10)–(5.13). Since ℚ is a subspace it follows that ⃗ − s(P⃗ + Q)⃗ : M⃗ = sp(P), ⃗ p(sP)⃗ = inf Ξ(s(P⃗ + Q)) 0 ⃗ Q∈ℚ

where s ≥ 0. For any ϵ > 0 there exists Q⃗ 1 , Q⃗ 2 ∈ ℚ such that p(P⃗ 1 ) ≤ Ξ(P⃗ 1 + Q⃗ 1 ) − (P⃗ 1 + Q⃗ 1 ) : M⃗ 0 + ϵ, p(P⃗ 2 ) ≤ Ξ(P⃗ 2 + Q⃗ 2 ) − (P⃗ 2 + Q⃗ 2 ) : M⃗ 0 + ϵ, thus, by the sublinearity of Ξ and the definition of p, p(P⃗ 1 + P⃗ 2 ) ≤ Ξ(P⃗ 1 + P⃗ 2 + Q⃗ 1 + Q⃗ 2 ) − (P⃗ 1 + P⃗ 2 + Q⃗ 1 + Q⃗ 2 ) : M⃗ 0 ≤ Ξ(P⃗ 1 + Q⃗ 1 ) − (P⃗ 1 + Q⃗ 1 ) : M⃗ 0 + Ξ(P⃗ 2 + Q⃗ 2 ) − (P⃗ 2 + Q⃗ 2 ) : M⃗ 0 ≤ p(P⃗ 1 ) + p(P⃗ 2 ) + 2ϵ,

(7.9)

so p is sublinear on 𝕄󸀠 (N, J). Let U be the one-dimensional space of 𝕄󸀠 (N, J) spanned by P⃗ 0 . Define f (sP⃗ 0 ) := sp(P⃗ 0 ) for any s ∈ ℝ. Thus, f is a linear functional on U and satisfies f (P)⃗ ≤ p(P)⃗ for any P⃗ ∈ U. Indeed, it holds with quality if P⃗ = sP⃗ 0 , where s ≥ 0 by definition, while f (P)⃗ ≤ 0 ≤ p(P)⃗ if s ≤ 0 by (7.8). By the Hahn–Banach theorem there exists a linear functional F ≡ M⃗ ∗ ∈ 𝕄+ (N, J) such that P⃗ : M⃗ ∗ ≤ p(P)⃗ for any P⃗ ∈ 𝕄󸀠 (N, J) while P⃗ 0 : M⃗ ∗ = p(P⃗ 0 ). Thus P⃗ : M⃗ ∗ ≤ Ξ(P⃗ + Q)⃗ − (P⃗ + Q)⃗ : M⃗ 0 holds for any P⃗ ∈ 𝕄󸀠 (N, J) and any Q⃗ ∈ ℚ. Thus (P⃗ + Q)⃗ : (M⃗ ∗ + M⃗ 0 ) ≤ Ξ(P⃗ + Q)⃗ + Q⃗ : M⃗ ∗

7.1 Optimality within the weak partitions | 87

holds for any P⃗ ∈ 𝕄󸀠 (N, J) and Q⃗ ∈ ℚ. Setting Q⃗ = 0 we obtain that M⃗ ∗ + M⃗ 0 ∈ S by Theorem 5.1, and setting P⃗ = −Q⃗ we obtain Q⃗ : M⃗ ∗ ≥ 0 on ℚ. Since ℚ is a subspace it follows that Q⃗ : M⃗ ∗ = 0 for any Q⃗ ∈ ℚ, so M⃗ ∗ ∈ ℚ⊥ . We obtain that sup

⊥ +M⃗ )∩S ⃗ M∈(ℚ 0

P⃗ 0 : M⃗ ≥ P⃗ 0 : M⃗ ∗ = inf Ξ(P⃗ 0 + Q)⃗ − Q⃗ : M⃗ 0 . ⃗ Q∈ℚ

This implies the opposite inequality to (7.7). 7.1.2 Optimal multi(sub)partitions: extended setting Given ζ ̄ ∈ C(X, ℝJ+ ) as in (5.1) and θ⃗ ∈ C(X, ℝN ) as in Assumption 4.0.1, we consider the function ζ̂ := (ζ1 , . . . ,ζJ+N ) := (ζ ̄ , θ)⃗ ∈ C(X, ℝN+J ). This definition suggests that we extend the set of “goods” from 𝒥 to 𝒥 ∪ ℐ . Thus, we ̂󸀠 := 𝕄󸀠 (N, J) × 𝕄󸀠 (N, N), where 𝕄󸀠 (N, J) as in Definiconsider the extended spaces 𝕄 󸀠 N2 tion 5.2.1 and 𝕄 (N, N) ∼ ℝ parameterized by 𝕄󸀠 (N, N) = (p⃗ ∗,1 , . . . , p⃗ ∗,N ), p⃗ ∗,i ∈ ℝN . This space is parameterized as ̂ := (P,⃗ P⃗ ∗ ) = (p⃗ 1 , . . . , p⃗ N ; p⃗ ∗,1 , . . . , p⃗ ∗,N ) ∼ ℝN(N+J) P (recall p⃗ i ∈ ℝJ and p⃗ ∗,i ∈ ℝN for 1 ≤ i ≤ N). ̂+ := 𝕄+ (N, J) × 𝕄+ (N, N), thus Similarly, the dual space 𝕄 ̂ := (M,⃗ M⃗ 󸀠 ) = (m⃗ 1 , . . . , m⃗ N ; m⃗ ∗,1 , . . . , m⃗ ∗,N ) ∼ ℝN(N+J) M as well. The duality action of (P,⃗ P⃗ 󸀠 ) on (M,⃗ M⃗ ∗ ) is the direct sum ̂:M ̂ := P⃗ : M⃗ + P⃗ 󸀠 : M⃗ 󸀠 := ∑ (p⃗ i ⋅ m⃗ i ) J + ∑ (p⃗ ∗,i ⋅ m⃗ ∗,i ) N , P ℝ ℝ i∈ℐ

i∈ℐ

where the inner products refer to the corresponding spaces indicated for clarity. Let ̂0 (P,⃗ P⃗ ) := μ (max (p⃗ ⋅ ζ ̄ + p⃗ ⋅ θ)) ⃗ Ξ ∗ i ∗,i ζ

(7.10)

̂ 0,+ ⃗ ⃗ Ξ (P, P∗ ) := μ (max (p⃗ i ⋅ ζ ̄ + p⃗ ∗,i ⋅ θ,⃗ ) ∨ 0) . ζ

(7.11)

i∈ℐ

resp. i∈ℐ

Comparing with (5.10)–(5.12) we observe that (7.10), (7.11) are just the applications of these definitions to the current setting.

88 | 7 Optimal multipartitions Definition 7.1.1. i) ℚ := {(P,⃗ 0⃗ 𝕄󸀠 (N,N) ); P⃗ ∈ 𝕄󸀠 (N, J)}, then ℚ⊥ := {(0⃗ M⃗ , M⃗ ∗ ); M⃗ ∗ ∈ 𝕄+ (N, N)}. ̂ 0 := (0⃗ 𝕄󸀠 (N,J) , I󸀠 ), where I󸀠 is the identity N × N matrix. ii) P 0 0 ̂0 := (M⃗ 0 , 0⃗ 𝕄 (N,N) ), where M⃗ 0 ∈ 𝕄+ (N, J) is given. iii) M + With this notation we get (Definition 7.1.3 below) ⃗ ̂0 : μ⃗ (ζ̂) ≡ I󸀠 : μ(⃗ θ). θ(μ)⃗ ≡ ∑ μi (θi ) ≡ P 0

(7.12)

ξζθ (x, P)⃗ := max {θi (x) + p⃗ i ⋅ ζ ̄ (x), 0} ,

(7.13)

ξζθ,+ (x, P)⃗ := ξζθ (x, P)⃗ ∨ 0.

(7.14)

i∈ℐ

Let i∈ℐ

Then, (7.10), (7.11) can be written as ̂ ⃗ 󸀠 θ,+ ⃗ ⃗ (P, I0 ) = Ξθ,+ Ξ0,+ ζ (P) := μ(ξζ (⋅, P)), ζ

(7.15)

̂0 (P,⃗ I󸀠 ) = Ξθ (P)⃗ := μ(ξ θ (⋅, P)). ⃗ Ξ 0 ζ ζ ζ

(7.16)

Proposition 7.1 can now be written as follows. ̄ the maximum of θ(μ)⃗ in 𝒫 w ⃗ (resp. Theorem 7.3. Given M⃗ ∈ ΔN (μ)̄ (resp. M⃗ ∈ ΔN (μ)), {M } 0

the maximum of θ(μ)⃗ in 𝒮𝒫 w ) is given by {M⃗ } 0

+

Σθζ (M⃗ 0 ) =

inf

⃗ ⃗ ⃗ [Ξθ,+ ζ (P) − P : M0 ] ,

(7.17)

inf

[Ξθζ (P)⃗ − P⃗ : M⃗ 0 ] .

(7.18)

󸀠 (N,J) ⃗ P∈𝕄

resp. Σθζ (M⃗ 0 ) =

󸀠 (N,J) ⃗ P∈𝕄

In Theorem 7.3 we left open the question of existence of a minimizer P⃗ of (7.17), (7.18). See Theorem 7.4 below. Definition 7.1.2. M⃗ ∈ ΔN (μ)̄ is an escalating capacity if there is no P⃗ minimizing (7.18). The reason for this notation will be explained in Section 7.1.3. See also the box above Theorem 7.5 and Section 7.2.1. Definition 7.1.3. Given a weak (sub)partition μ.⃗ Let M⃗ ∗ (μ)⃗ ∈ 𝕄+ 󸀠 (N, N) given by {μ(j) (θi )}1≤i,j≤N . The extended feasibility set is an extension of Definition 5.1.2 ⃗ μ), ? ̄ := ⋃ {M( ⃗ M⃗ ∗ (μ)} ⃗ ; Δ N (μ) w ⃗ μ∈𝒫

? ̄ := Δ N (μ)

⃗ μ), ⃗ M⃗ ∗ (μ)} ⃗ , ⋃ {M(

w ⃗ μ∈𝒮𝒫

7.1 Optimality within the weak partitions | 89

and ⃗ , ΔN (μ)̄ 󸀠 := ⋃ {M⃗ ∗ (μ)} w ⃗ μ∈𝒫

resp. ΔN (μ)̄ 󸀠 :=

⃗ . ⋃ {M⃗ ∗ (μ)}

w ⃗ μ∈𝒮𝒫

The diagonal elements of M⃗ ∗ (μ)⃗ are called the surplus values of the agents under the (sub)partition μ:⃗ ⃗ ≡ (μ1 (θ1 ), . . . , μN (θN )) , Diag (M⃗ ∗ (μ)) where μi (θi ) is the surplus value of agent i. Consistently with Definition 7.1.1 and (7.5), (7.6), we define ? ? ̄ := {(M,⃗ t) ∈ 𝕄+ (N, J) ⊗ ℝ; (M,⃗ M⃗ ∗ ) ∈ Δ ̄ t = Tr (M⃗ ∗ )} , Δ N (μ) N (μ); resp. ? ? ̄ := {(M,⃗ t) ∈ 𝕄+ (N, J) ⊗ ℝ; (M,⃗ M⃗ ∗ ) ∈ Δ ̄ t = Tr(M⃗ ∗ )} . Δ N (μ) N (μ); Note: In terms of this definition, as well as Definition 7.1.1(ii), ⃗ θ(μ)⃗ ≡ I󸀠0 : M⃗ ∗ (μ)⃗ ≡ Tr (M⃗ ∗ (μ)) is another equivalent formulation of (7.12). In particular, Theorem 7.3 implies the following, alternative definition for the optimal value of θ(μ)⃗ on 𝒫{wM}⃗ (resp. on 𝒮𝒫 w ⃗ ): {M} ⃗ := Σθζ (M)

sup

? ̄ Tr(M⃗ ∗ ) ≡ sup {t; (M,⃗ t) ∈ Δ N (μ)}

(7.19)

sup

? ̄ . Tr(M⃗ ∗ ) ≡ sup {t; (M,⃗ t) ∈ Δ N (μ)}

(7.20)

̄ (M,⃗ M⃗ ∗ )∈Δ? N (μ)

resp. +

⃗ := Σθζ (M)

̄ (M,⃗ M⃗ ∗ )∈Δ? N (μ) +

⃗ ∈ 𝜕Δ ? ̄ for any M⃗ ∈ ΔN (μ). ̄ It is also evident From (7.20) we obtain that (M,⃗ Σθζ (M)) N (μ) θ ⃗ ∈ 𝜕Δ ? ? ̄ since ΔN (μ)̄ (hence Δ ̄ contains no interior points. We that (M,⃗ Σ (M)) N (μ), N (μ)) ζ

now imply Corollary 5.2.1 to obtain the following.

? ̄ if and only if (P,⃗ s) = 0 is the only miniCorollary 7.1.1. (M,⃗ t) is an inner point of Δ N (μ) mizer of inf

󸀠 (N,J)⊕ℝ) ⃗ (P,s)∈(𝕄

̂ ⃗ 󸀠 Ξ0,+ (P, sI0 ) − (P,⃗ s) : (M,⃗ t) = 0. ζ

90 | 7 Optimal multipartitions ̄ there exist (P,⃗ s) ≠ 0 such that Hence, for any M⃗ ∈ ΔN (μ)̄ (resp. M⃗ ∈ ΔN (μ)) + ̂ 0,+ ⃗ Ξ (P,⃗ sI󸀠0 ) = P⃗ : M⃗ + sΣθζ (M) ζ

(7.21)

̂0 (P,⃗ sI󸀠 ) = P⃗ : M⃗ + sΣθ (M). ⃗ Ξ 0 ζ ζ

(7.22)

resp.

To understand the meaning of (7.21), (7.22) we compare it to Theorem 7.3. By (7.13)– (7.16) we may write ̂ ⃗ 󸀠 ̂0 ⃗ 󸀠 sθ ⃗ ⃗ (P, sI0 ) = Ξsθ,+ Ξ0,+ ξ (P) resp. Ξζ (P, sI0 ) = Ξξ (P), ζ so (7.21), (7.22) are equivalent to the following. ̄ there exists (P⃗ 0 , s0 ) ≠ 0 such that Theorem 7.4. For any M⃗ ∈ ΔN (μ)̄ (resp. M⃗ ∈ ΔN (μ)) inf

󸀠 (N,J)×ℝ ⃗ (P,s)∈𝕄

+

s θ,+

θ 0 ⃗ ⃗ ⃗ ⃗ [Ξsθ,+ ζ (P) − sΣζ (M) − P : M] = Ξζ

+

⃗ − P⃗ 0 : M⃗ = 0, (P⃗ 0 ) − s0 Σθζ (M) (7.23)

resp. inf

󸀠 (N,J)×ℝ ⃗ (P,s)∈𝕄

s θ

θ ⃗ θ ⃗ 0 ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ [Ξsθ ζ (P) − sΣζ (M) − P : M] = Ξζ (P0 ) − s0 Σζ (M) − P0 : M = 0.

(7.24)

Now: (P⃗ 0 , s0 ) ≠ 0 implies that either s0 ≠ 0 or P⃗ 0 ≠ 0 (or both). If s0 ≠ 0 (in that case the reader can show that, in fact, s0 > 0) we can divide (7.23), (7.24) by s0 , using ⃗ (7.13)–(7.16) to observe s−1 Ξsθ (P)⃗ = Ξθ (P/s), and conclude that there exists a minimizer ζ

⃗ s−1 0 P0 to (7.17), (7.18) in Theorem 7.4. In particular:

ζ

The case of escalation (Definition 7.1.2) corresponds to s0 = 0 (hence P⃗ 0 ≠ 0) in Theorem 7.4.

The theorem below implies another characterization of the optimal (sub)partition. Theorem 7.5. Any optimal (sub)partition μ⃗ corresponding to M⃗ ∈ ΔN (μ)̄ (resp. M⃗ ∈ ̄ satisfies the following: ΔN (μ)) supp(μi ) ⊂ Aθ,+ := {x ∈ X; p⃗ 0,i ⋅ ζ ̄ (x) + s0 θi (x) = max [p⃗ 0,k ⋅ ζ ̄ (x) + s0 θk (x)]+ } , (7.25) i k∈ℐ

resp. supp(μi ) ⊂ Aθi := {x ∈ X; p⃗ 0,i ⋅ ζ ̄ (x) + s0 θi (x) = max p⃗ 0,k ⋅ ζ ̄ (x) + s0 θk (x)} , k∈ℐ

where P⃗ 0 = (p⃗ 0,1 , . . . , p⃗ 0,N ) and s0 ∈ ℝ are as given by Theorem 7.4.

(7.26)

7.1 Optimality within the weak partitions | 91

Proof. Let μ⃗ := (μ1 , . . . , μN ) be an optimal (sub)partition and let μ0 = μ − ∑i∈ℐ μi . By (7.23), s θ,+

0 = Ξζ 0

+

⃗ − P⃗ 0 : M,⃗ (P⃗ 0 ) − s0 Σθζ (M)

while by (7.16) and since μ⃗ ∈ 𝒮𝒫 w ⃗ is an optimal (sub)partition {M} s θ,+

Ξζ 0

(P⃗ 0 ) =

∑ μi (max ([s0 θk + p⃗ 0,k ⋅ ζ ̄ ]+ )) k∈ℐ

i∈ℐ∪{0}

N

+ ⃗ + P⃗ 0 : M,⃗ ≥ ∑ [s0 μi (θi ) + μi (p⃗ 0,i ⋅ ζ ̄ )] = s0 Σθζ (M) i=1

(7.27)

so the inequality above is an equality. In particular, for μi -a. e. max[s0 θk (x) + p⃗ 0,k ⋅ ζ ̄ (x)]+ = s0 θi (x) + p⃗ 0,i ⋅ ζ ̄ (x). k∈ℐ

By (7.25) and the continuity of θ,⃗ ζ ̄ we obtain that supp(μi ) ⊂ Aθ,+ . i ⃗ The case M ∈ ΔN (μ)̄ is proved similarly.

7.1.3 Price adaptation and escalation So far we considered the equilibrium vector p⃗ as a tool for achieving optimal (sub)partitions (Sections 4.2 and 4.3, see also Proposition 4.4). One may expect that, in the (j) case of multipartition, the price −pi should be interpreted as the equilibrium price (j)

charged by agent i for the good j in order to obtain the required capacity m0,i . However, we did not consider how the agent determines these prices. It is conceivable that this process is conducted by trial and error. Thus, when agent i “guesses” the (j) price vector −pi for the good j ∈ 𝒥 , she should consider the number of consumers of (j)

j who accept this price and compare it with the desired capacities m0,i . If she is un(j)

derbooked, namely, m0,i is above the number of her consumers for j, she will decrease the price in order to attract more consumers. If, on the other hand, she is overbooked, then she will increase the price to get rid of some. But how does agent i determine the number of consumers of j who accept the (j) price −pi ? Recall that each consumer x needs the fraction ζj (x) of the good j. Hence the price paid by consumer x to agent i for the basket J is −ζ ̄ (x) ⋅ p⃗ i . Thus, she only needs to determine the entire set of her consumers μi . Once μi is known, she knows (j) the current capacity mi (P)⃗ = μi (ζj ) for the current price matrix −P.⃗ Recalling (4.14) we obtain that the set of all candidates A+ (P)⃗ who may hire i at i

the price level −p⃗ i is the set of consumers who make a non-negative profit for trading

92 | 7 Optimal multipartitions with i, and this profit is at least as large as the profit they may get from trading with any other agent. Thus A+i (P)⃗ := {x ∈ X; θi (x) + p⃗ i ⋅ ζ ̄ ≥ [θk (x) + p⃗ k ⋅ ζ ̄ (x)]+ ∀k ∈ ℐ }.

(7.28)

In fact, there may be a set of “floating” consumers who belong to two (or more) such ⃗ is not necessarily zero for i ≠ k). The only information sets (note that μ(A+i (P)⃗ ∩ A+k (P)) which i can be sure of, upon her choice of the price vector −p⃗ i , is that all her consumers ⃗ are in the set A+i (P). +

Note that Ξθζ and Ξθζ are convex functions. By Proposition A.9 (recalling (7.16) and ⃗ P)⃗ Definition A.5.1) we get that, under the choice P,⃗ the corresponding capacity set M( is given by the subgradient

+

⃗ P)⃗ ∈ 𝜕 ⃗ Ξθ ≠ 0. M( −P ζ

(7.29)

(j)

⃗ P). ⃗ According to the above reasoning, Let mi (P)⃗ be in the (i, j)-component of M( (j)

the agent i will decrease pi if

(j) (j) m0,i > max mi (P)⃗ ⃗ M( ⃗ P)⃗ M∈

(j)

and increase pi if (j)

(j)

⃗ m0,i < min mi (P). ⃗ M( ⃗ P)⃗ M∈

If (j) ⃗ ⃗ P)} ⃗ ≤ m(j) ≤ max {m(j) (P); ⃗ M⃗ ∈ M( ⃗ P)} ⃗ , min {mi (P); M⃗ ∈ M( 0,i i (j)

then i will, probably, not change pi . ⃗ is the value of the price matrix at time t and P⃗ 0 is its initial value at So, if −P(t) ⃗ t = t0 we presume that the forward derivative d+ P(t)/dt exists and would like to state + ⃗ + ⃗ ⃗ ⃗ ⃗ that d P(t)/dt ∈ M(P(t)) − M0 . However, d P(t)/dt, if it exists, is in the space 𝕄󸀠 (N, J), ⃗ P)⃗ and M⃗ 0 are in 𝕄+ (N, J). So, we have to identify 𝕄󸀠 (N, J) and 𝕄+ (N, J) in while M( some way. For this we define a linear mapping J : 𝕄+ (N, J) → 𝕄󸀠 (N, J) such that JM⃗ : M⃗ > 0 ∀M⃗ ≠ 0

in 𝕄+ (N, J).

This definition makes 𝕄+ (N, J) and 𝕄󸀠 (N, J) inner product space, and |M|⃗ := √JM⃗ : M;⃗ |P|⃗ := √P⃗ : J−1 P⃗ are natural norms.

(7.30)

7.1 Optimality within the weak partitions | 93

So, we presume that d+ ⃗ ⃗ P(t)) ⃗ P(t) ∈ J (M( − M⃗ 0 ) , t ≥ t0 ; dt

⃗ 0 ) = P⃗ ∈ 𝕄󸀠 (N, J). P(t

(7.31)

The condition (7.31) is an example of a differential inclusion. It is a generalization of a system of ordinary differential equations (ODEs). In fact, by (7.29) we observe that it + is an ODE if the subgradient of Ξθζ is a singleton, which is equivalent, via Proposi+

tion A.10, to the assumption that Ξθζ is differentiable anywhere. ⃗ satisfying (7.31) is common knowledge, due The existence and uniqueness of P(⋅) +

the convexity of Ξθζ ([2], [3]). For the sake of completeness we introduce below some of the steps toward the proof of this result. ⃗ j ) is known, define Let ϵ > 0 and tj := t0 + jϵ. If P(t θ+ −2 󵄨󵄨 ⃗ ⃗ 󵄨󵄨󵄨2 ⃗ ⃗ ⃗ ⃗ j+1 ) := min! ⃗ 󸀠 󵄨 P(t P∈𝕄 (N,J) {(2ϵ) 󵄨󵄨P(tj ) − P󵄨󵄨 + Ξζ (−P) + P : M0 } .

(7.32)

+

⃗ j+1 ) is Since P⃗ → Ξθζ (−P)⃗ is convex, the term in brackets above is strictly convex and P(t ⃗ unique (Definition A.2.1). Moreover, it follows from (7.32) and (7.29) that P(tj+1 ) satisfies the implicit inclusion

⃗ j+1 ) ∈ P(t ⃗ j ) + ϵJ (M( ⃗ P(t ⃗ j+1 )) − M⃗ 0 ) . P(t

(7.33)

Next, we interpolate on time to define t → P⃗ ϵ (t) for any t ≥ t0 as ⃗ j+1 ) + (tj+1 − t)P(t ⃗ j )] P⃗ ϵ (t) = ϵ−1 [(t − tj )P(t

for tj ≤ t < tj+1 , j = 0, 1, . . . .

⃗ = limϵ→0 P⃗ ϵ (t) for any t ≥ t0 is the unique soluUsing (7.33) it can be proved that P(t) tion of the inclusion (7.31). It is also evident from (7.32) that +

+

Ξθζ (−P⃗ tj+1 ) + P⃗ tj+1 : M⃗ 0 ≤ Ξθζ (−P⃗ tj ) + P⃗ tj : M⃗ 0 for any j = 0, 1, 2 . . .. Hence +

⃗ ⃗ : M⃗ 0 t → Ξθζ (−P(t)) + P(t) is non-increasing. Moreover, by (7.29), (7.30), (7.31) d+ θ + ⃗ θ+ ⃗ : M⃗ 0 ] = −J (d+ P(t)/dt) ⃗ ⃗ [Ξζ (−P(t)) + P(t) : (𝜕−P(t) ⃗ Ξζ − M0 ) dt 󵄨 ⃗ ⃗ 󵄨2 = − 󵄨󵄨󵄨󵄨(M( P(t)) − M⃗ 0 )󵄨󵄨󵄨󵄨 .

(7.34)

Recall from Theorem 7.3 that +

Σθζ (M⃗ 0 ) =

inf

󸀠 (N,J) ⃗ P∈𝕄

θ,+ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ [Ξθ,+ ζ (P) − P : M0 ] ≤ Ξζ (P(t)) − P(t) : M0

for any t ≥ t0 . We now obtain the reason for the terminology of “escalating capacity” in Definition 7.1.2.

94 | 7 Optimal multipartitions Theorem 7.6. The solution of (7.31) satisfies +

θ ⃗ ⃗ ⃗ ⃗ lim Ξθ,+ ζ (P(t)) − P(t) : M0 = Σζ (M0 ).

t↑∞

⃗ = P⃗ 0 , where P⃗ 0 is a minimizer of (7.17). OtherIf M⃗ 0 is non-escalating, then limt↑∞ P(t) ⃗ ⃗ wise, the limit of P(t) does not exist and limt↑∞ |P(t)| = ∞. However, ⃗ P(t) := P⃗ 0 ⃗ t↑∞ |P(t)| lim

exists, where P⃗ 0 is a minimizer of (7.23).1

7.2 Optimal strong multipartitions Recall the definition of θ on the set of strong N-(sub)partitions: N

θ(A)⃗ := ∑ ∫ θi dμ. i=1 A

i

Let 𝒦 ⊂ 𝕄+ (N, J) be a compact convex set. The main question we address in this section is the following. Under which conditions is there a unique, strong (sub)partition A⃗ which maximizes θ in the set of w,ζ

w,ζ ̄

weak (sub)partitions 𝒫{𝒦} (resp. 𝒮𝒫 {𝒦} )?

Following the discussion of Chapter 4 and Theorems 7.4 and 7.5, we focus on the “natural suspects” Aθi (P)⃗ := {x ∈ X; p⃗ i ⋅ ζ ̄ (x) + θi (x) > max p⃗ j ⋅ ζ ̄ (x) + θj (x)} ,

(7.35)

θ ⃗ θ ⃗ ⃗ Aθ,+ i (P) := Ai (P) − A0 (P),

(7.36)

j=i̸

where Aθ0 (P)⃗ := {x ∈ X; p⃗ i ⋅ ζ ̄ (x) + θi (x) ≤ 0 ∀i ∈ ℐ } . Recall that the utility of a consumer x of agent i charging price p⃗ i is θi (x) − p⃗ i ⋅ ζ ̄ (x). ⃗ represents subsets of consumers who Thus, the set of (sub)partitions Aθi (−P)⃗ (Aθ,+ (−P)) i prefer agent i over all other agents, given the price matrix P.⃗ As suggested by Theorem 7.4, there is a close relation between optimal μ⃗ and strong (sub)partitions of the form (7.35), (7.36). Thus we rephrase our question as follows: 1 Note that in the case of escalating M⃗ 0 , s0 = 0 while P⃗ 0 ≠ 0 in (7.23).

7.2 Optimal strong multipartitions | 95

Under which conditions is there a unique P⃗ ∈ 𝕄󸀠 (N, J) such that (7.35) (resp. (7.36)) are θ-optimal w,ζ ̄

w,ζ

strong (sub)partitions in 𝒫{𝒦} (resp. 𝒮𝒫 {𝒦} )?

⃗ is a singleton. Recall from At the first stage we concentrate on the case where 𝒦 = {M} (7.17), (7.18) that +

⃗ = Σθζ (M)

inf

⃗ ⃗ ⃗ [Ξθ,+ ζ (P) − P : M] ,

(7.37)

inf

⃗ , [Ξθζ (P)⃗ − P⃗ : M]

(7.38)

󸀠 (N,J) ⃗ P∈𝕄

Σθζ (M⃗ 0 ) =

󸀠 (N,J) ⃗ P∈𝕄

where, from (7.15), (7.16), θ,+ ⃗ ⃗ Ξθ,+ ζ (P) ≡ μ (ξζ (x, p)) ;

⃗ ; Ξθζ (P)⃗ ≡ μ (ξζθ (x, P))

ξζθ,+ (x, P)⃗ ≡ max(θi (x) + p⃗ i ⋅ ζ ̄ (x))+ ,

(7.39)

ξζθ (x, P)⃗ ≡ max(θi (x) + p⃗ i ⋅ ζ ̄ (x)).

(7.40)

i

i

We now consider the following adaptation of Assumption 6.2.1. Assumption 7.2.1. i) For any i ≠ j ∈ ℐ and any p⃗ ∈ ℝJ , μ(x ∈ X; p⃗ ⋅ ζ ̄ (x) + θi (x) − θj (x) = 0) = 0. ii) For any i ∈ ℐ and any p⃗ ∈ ℝJ , μ(x ∈ X; θi (x) = p⃗ ⋅ ζ ̄ (x)) = 0. ⃗ is, indeed, a strong partition for any By Assumption 7.2.1(i) it follows that {Aθi (P)} 󸀠 ⃗ ⃗ is a strong subP ∈ 𝕄 (N, J). Likewise, Assumption 7.2.1(i), (ii) implies that {Aθ,+ (P)} i partition. In particular, θ,+ ⃗ ⃗ Aθi (P)⃗ ∩ Aθj (P)⃗ = Aθ,+ i (P) ∩ Aj (P) = 0

(7.41)

for i ≠ j. 7.2.1 Example of escalation Let P⃗ 0 , M⃗ 0 be given as in Corollary 6.2.2. Then (A1 (P⃗ 0 ), . . . , AN (P⃗ 0 )) given by (6.9) where P⃗ 0 is substituted for P⃗ is the only partition in 𝒫 ζ {M⃗ } . Assume that θi −θj is independent 0 of the components of ζ ̄ on Ā i (P⃗ 0 ) ∩ Ā j (P⃗ 0 ).2 This implies that for any (λ1 , . . . , λN ) ∈ ℝN there exists x ∈ Ā (P⃗ ) ∩ Ā (P⃗ ) such that θ (x) − θ (x) ≠ λ⃗ ⋅ ζ ̄ (x). i

0

j

0

2 Ā stands for the closure of the set A.

i

j

96 | 7 Optimal multipartitions Proposition 7.2. For M⃗ 0 and θ as above, M⃗ 0 is an escalating capacity for the given θ. ζ Proof. If a minimizer P⃗ = (p⃗ 1 . . . p⃗ N ) of (7.18) exists, then (7.35) is a partition in 𝒮𝒫

which by Corollary 6.2.2 must be the same as Ai (P⃗ 0 ). In particular,

{M⃗ 0 }

⃗ Ā i (P⃗ 0 ) ∩ Ā j (P⃗ 0 ) = Ā θi (P)⃗ ∩ Ā θj (P). Any point in the set Ā θi (P)⃗ ∩ Ā θj (P)⃗ must satisfy θi (x) + p⃗ i ⋅ ζ ̄ (x) = θj (x) + p⃗ j ⋅ ζ ̄ (x), so θi (x) − θj (x) = (p⃗ j − p⃗ i ) ⋅ ζ ̄ (x). Since any such point is in Ā i (P⃗ 0 ) ∩ Ā j (P⃗ 0 ) as well, we obtain a contradiction to the assumption on θ. 7.2.2 Uniqueness for a prescribed capacity It turns out that Assumption 7.2.1, standing alone, is enough for the uniqueness of the ̄ provided M⃗ is an interior point of ΔN (μ). ̄ The key optimal subpartition for M⃗ ∈ ΔN (μ), to this result is the following observation, generalizing Lemma 6.3. Lemma 7.1. Under Assumption 7.2.1(i), Ξθζ is differentiable at any P⃗ ∈ 𝕄󸀠 (N, J) and satisfies (7.42)(a). If, in addition, Assumption 7.2.1(ii) is satisfied, then Ξθ,+ is differentiable ζ as well and (7.42b) holds. Here a)

𝜕Ξθζ 𝜕p⃗ i

(P)⃗ = ∫ ζ ̄ dμ,

b)

Aθi (p)⃗

𝜕Ξθ,+ ζ 𝜕p⃗ i

∫ ζ ̄ dμ.

(P)⃗ =

(7.42)

Aθ,+ (p)⃗ i

Remark 7.2.1. Note that the conditions of Lemma 6.3 as well as Assumption 6.2.1 are not required in Lemma 7.1. ̄ then Theorem 7.7. Consider Assumption 7.2.1(i), (ii). If M⃗ is an interior point of ΔN (μ),

there exists a unique subpartition which maximizes θ in 𝒮𝒫 a strong one, given by

{Aθ,+ (P⃗ 0 )} i

w,ζ ̄ , ⃗ {M}

and this subpartition is

(7.36) for some uniquely determined P⃗ 0 ∈ 𝕄󸀠 (N, J).

̄ = 0, so Theorem 7.7 is void for M⃗ ∈ ΔN (μ). ̄ Note that int(ΔN (μ)) Proof. Let −Σθ,+ be the Legendre transforms of Ξθ,+ (−⋅), i. e., ζ ζ ⃗ Σθ,+ ζ (M) =

inf

󸀠 (N,J) ⃗ P∈𝕄

⃗ ⃗ ⃗ Ξθ,+ ζ (P) − P : M.

(7.43)

We prove that the essential domain of −Σθ,+ is the same as the essential domain of ζ ̄ Indeed, by definition (7.39), (5.13), , namely, ΔN (μ). −Σ0,+ ζ

θ,+ ⃗ 0,+ ⃗ ⃗ ⃗ ⃗ Ξ0,+ ζ (P) + ‖θ‖∞ μ(X) ≥ Ξζ (P) ≥ Ξζ (P) − ‖θ‖∞ μ(X),

7.2 Optimal strong multipartitions | 97

and hence inf

󸀠 (N,J) ⃗ P∈𝕄

⃗ ⃗ ⃗ Ξθ,+ ζ (P) − P : M > −∞ ⇔

inf

󸀠 (N,J) ⃗ P∈𝕄

⃗ ⃗ ⃗ Ξ0,+ ζ (P) − P : M > −∞,

which implies the claim via Theorem 5.1. If M⃗ is an interior point in the essential domain of −Σθ,+ , then by (A.5.1) the subζ θ,+ θ,+ ⃗ gradient 𝜕 ⃗ (−Σ ) is not empty. Any P ∈ −𝜕 ⃗ (−Σ ) is a minimizer of (7.43). Let P⃗ 0 be M

M

ζ

ζ

such a minimizer. Let μ⃗ be a maximizer of θ in 𝒮𝒫

w,ζ ̄ . ⃗ {M}

Then by (7.39)

θ,+ ⃗ 0 θ,+ ⃗ ⃗0 ⃗ ⃗0 ⃗0 ⃗ Σθ,+ ζ (M) = Ξζ (P ) − P ⋅ M = μ(ξζ (x, P )) − P ⋅ M

≥ ∑ μi (ξζθ,+ (⋅, P⃗ 0 )) − P⃗ 0 ⋅ M⃗ ≥ ∑ μi (θi + p⃗ 0i ⋅ ζ ̄ ) − P⃗ 0 ⋅ M.⃗ i∈ℐ

i∈ℐ

(7.44)

By Lemma 7.1 we obtain that Ξθ,+ is differentiable at P⃗ 0 and by (7.42)3 ζ μi (ζ ̄ ) =

∫

ζ ̄ dμ = m⃗ i .

Aθ,+ (P⃗ 0 ) i

Hence ⃗ (7.44) = ∑ μi (θi ) ≡ θ(μ)⃗ = Σθ,+ ζ (M), i∈ℐ

where the last equality follows from Theorem 7.3. Hence the middle inequality in (7.44) is an equality. Since ξζθ,+ (x, P)⃗ ≥ θi + p⃗ i ⋅ ζ ̄ everywhere by (7.39), we obtain ξζθ,+ (x, p⃗ 0 ) = (P⃗ 0 ) by (7.36). This, (7.41), θi (x) + p⃗ 0i ⋅ ζ ̄ (x) for any x ∈ supp(μi ). That is, supp(μi ) ⊇ Aθ,+ i and μi ≤ μ imply that μi is the restriction of μ to Aθ,+ (P⃗ 0 ). In particular, the maximizer i

w,ζ ̄ μ⃗ ∈ 𝒮𝒫 ⃗ is unique, and is a strong subpartition given by A⃗ θ,+ (P⃗ 0 ). {M}

As a byproduct of the uniqueness of the minimizers P⃗ 0 of (7.43) and via Proposition A.10 we obtain the following. ̄ Corollary 7.2.1. Σθ,+ is differentiable at any inner point of its essential domain ΔN (μ). ζ Combining both Theorems 6.2 and 7.7 we obtain the following. ̄ then there exist Theorem 7.8. Consider Assumptions 7.2.1(i), (ii) and 6.2.1. If M⃗ ∈ ΔN (μ), a unique, maximal coalition ensemble D and a strong subpartition A⃗ D such that any w,ζ ̄ w,ζ ̄ μ⃗ ∈ 𝒮𝒫 which maximizes θ in 𝒮𝒫 is embedded in A⃗ . ⃗ {M}

⃗ {M}

D

3 Here is the only place in the proof we use the differentiability of Ξθ,+ . ζ

98 | 7 Optimal multipartitions ̄ then Theorem 7.7 implies the uniqueness of Proof. If M⃗ is an interior point of ΔN (μ), the subpartition corresponding to the coalition ensemble of individuals, which is the maximal possible coalition. ̄ then Theorem 6.2 implies the uniqueness of the maximal coalition If M⃗ ∈ 𝜕ΔN (μ), ⃗ Evidently, ensemble D and a unique strong subpartition A⃗ D corresponding to D(M). w,ζ ̄ must be embedded in A⃗ D . any subpartition in 𝒮𝒫 ⃗ {M}

If each agent {i} agrees on a fixed exchange rate z⃗(i) subject to Assumption 6.2.2, then we can get unconditional uniqueness. In fact, we have the following theorem. Theorem 7.9. Under Assumptions 7.2.1, 6.2.1, and 6.2.2, we have the following. If m⃗ ∈ w,ζ ̄ ⃗ ̄ then there exists a unique subpartition which maximizes θ in ⋃M∈ Z(Δ ⃗ Z⃗ −1 (m) N (μ)), ⃗ 𝒮𝒫 ⃗ , {M}

and this subpartition is a strong one.

Proof. We may assume that all components of m⃗ are in ℝN++ for, otherwise, we restrict ourselves to a subset of ℐ on which the components of m⃗ are all positive, and note that all the assumptions of the theorem are valid also for the restricted system. ⃗ ̄ then by the above assumption and Theorem 6.3, there exists a If m⃗ ∈ 𝜕Z(Δ N (μ)), unique subpartition and there is nothing to prove. So, we assume m⃗ is an inner point ⃗ ̄ By Theorem 7.3 of Z(Δ N (μ)). ⃗ ⃗ ⃗ ⃗ ⃗ ̄ ⃗ sup {Σθ,+ ζ (M) ; Z(M) = m} = sup {θ(μ); μi (zi ⋅ ζ ) = mi } . μ⃗

M⃗

(7.45)

On the other hand, θ,+ ⃗ 󸀠 ⃗ ⃗ ⃗ ⃗ ⃗ −1 ⃗ ⃗ ⃗ ⃗ ⃗ sup {Σθ,+ ζ (M) ; Z(M) = m} = sup inf {Ξζ (P) − P : M; P ∈ 𝕄 (N, J), M ∈ Z (m)} . M⃗

P⃗

M⃗

(7.46)

⃗ = m,⃗ then by Lemma 6.2 In addition, if Z(⃗ M) θ,+ ⃗ ∗ θ,+ ⃗ ∗ ⃗ ⃗ ⃗ ⃗∗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ inf Ξθ,+ ζ (P) − P : M ≤ inf Ξζ (Z (q)) − Z (q) ⋅ M = inf Ξζ (Z (q)) − q ⋅ m. P⃗

⃗ J q∈ℝ

⃗ J q∈ℝ

(7.47)

⃗ ̄ (which is the essential domain of Ξθ,+ Since m⃗ is an inner point of Z(Δ ∘Z⃗ ∗ ) and Ξθ,+ N (μ)) ζ ζ is differentiable at any point, Lemma 7.1 and Proposition A.9 imply that the infimum of the right side of (7.47) is attained at some q⃗ 0 ∈ ℝJ and mj =

𝜕 θ,+ ⃗ ∗ Ξ ∘ Z (q⃗ 0 ) = 𝜕qj ζ

zj⃗ ⋅ ζ ̄ dμ.

∫ Aθ,+ (Z⃗ ∗ (q⃗ 0 )) j

However, ξζθ,+ (x, Z⃗ ∗ (q⃗ 0 )) = zj⃗ ⋅ ζ ̄ (x) for any x ∈ Aθ,+ (Z⃗ ∗ (q⃗ 0 )), and hence j ⃗∗ ⃗ ⃗ ⃗ inf Ξθ,+ ζ (Z (q)) − q ⋅ m = ∑

⃗ J q∈ℝ

i∈ℐ

∫ Aθ,+ (Z⃗ ∗ (q⃗ 0 )) j

⃗ μi (zi⃗ ⋅ ζ ̄ ) = mi } . (7.48) zj⃗ ⋅ ζ ̄ (x)dμ ≤ sup {θ(μ); μ⃗

7.2 Optimal strong multipartitions | 99

Equations (7.45)–(7.48) imply that {Aθ,+ (Z⃗ ∗ (q⃗ 0 ))} is an optimal strong subpartition of θ j in ⋃M∈ ⃗ Z⃗ −1 (m) ⃗ 𝒮𝒫

w,ζ ̄ . ⃗ {M}

The uniqueness of this partition is proved as in Theorem 7.7.

7.2.3 Uniqueness within the feasibility domain Let us recall the generalized definition of undersaturation (US), saturation (S), and oversaturation (OS) (5.9), (5.8), (5.7). Theorems 7.8 and 7.9 deal with the existence and ̄ uniqueness of a strong (sub)partition maximizing θ for each prescribed M⃗ ∈ ΔN (μ). Here we discuss the uniqueness of optimal (sub)partition within closed convex sets 𝒦 ⊂ 𝕄+ (N, J). ⃗ (7.43) is the maximal value of θ for subpartitions for a preRecall that Σθ,+ (M) ζ ⃗ ̄ If we look for a subpartition maximizing θ on 𝒫 ζ 𝒦 (6.2) then it must scribed M ∈ ΔN (μ). ζ belong to 𝒫 ⃗ , where M⃗ 0 is a maximizer of Σθ,+ on Δ (μ)̄ ∩ 𝒦. Granted the uniqueness M0

ζ

N

of a maximal subpartition of θ in 𝒫 ζ {M}⃗ for any M⃗ ∈ ΔN (μ)̄ we obtain the following:

The uniqueness of the θ-maximizer in 𝒫 ζ 𝒦 is equivalent to the uniqueness of the maximizer of Σθ,+ ζ on ΔN (μ)̄ ∩ 𝒦.

Assumption 7.2.2. All the components of θ⃗ are non-negative on X and |θ|⃗ := ∑i∈ℐ θi (x) > 0 for any x ∈ X. Proposition 7.3. Under Assumption 7.2.2, if 𝒦 ⊂ 𝕄+ (N, J) is closed, then any maximizer ̄ In particular, if of Σθ,+ on ΔN (μ)̄ ∩ 𝒦 is necessarily obtained at the boundary of 𝒦 ∩ ΔN (μ). ζ ̄ then any such maximizer is in 𝜕ΔN (μ). ̄ Moreover, in that case any maximizing 𝒦 ⊃ ΔN (μ), subpartition is a partition. Indeed, if M⃗ 0 ∈ ̸ ΔN (μ)̄ is such a maximizer, then there is a strong subpartition ⃗ A realizing the maximum of θ in 𝒫 ζ {M⃗ } . In that case there exists a measurable set 0 A = X − ⋃n A such that μ(A ) > 0. Since at least one of the components of θ⃗ is 0

i=1

i

0

positive it follows that ∫A θi dμ > 0, ∫A ζi dμ = ϵ for some i ∈ ℐ , A ⊂ A0 , and ϵ > 0 small enough. If M⃗ 0 is an internal point of ΔN (μ)̄ ∩ 𝒦, then M⃗ 0 + ϵe⃗i ∈ ΔN (μ)̄ ∩ 𝒦 and Σθ,+ (M⃗ 0 + ϵe⃗i ) > Σθ,+ (M⃗ 0 ), which is a contradiction. ζ

ζ

We now extend Theorem 7.9 for a convex K⃗ ⊂ ℝN .

Theorem 7.10. Let K⃗ ⊂ ℝN be a closed convex set. ⃗ ̄ ≠ 0 there exists a unique Under Assumptions 7.2.1, 6.2.1, and 6.2.2: If K⃗ ∩ Z(Δ N (μ)) subpartition in 𝒮𝒫

w,ζ ̄ Z⃗ −1 (K)

which maximizes θ, and this subpartition is a strong one.

⃗ ̄ and Assumption 7.2.2 is granted as well, then the above subpartition If K ⊃ Z(Δ N (μ)) is a partition. In the last case Assumption 7.2.1 can be replaced by Assumption 7.2.1(i).

100 | 7 Optimal multipartitions Proof of Theorem 7.10. By Theorem 7.9 we only have to prove the uniqueness of the maximizer of ⃗ N (μ)), ⃗ m⃗ ∈ K⃗ ∩ Z(Δ ̄ m⃗ 󳨃→ Σθζ (m), where ⃗ ; Z(⃗ M) ⃗ = m} ⃗ := sup {Σθζ (M) ⃗ . Σθζ (m) M⃗

Let m⃗ 0 be this maximizer. Then, by (7.45)–(7.47) ⃗ = inf Ξθζ (Z⃗ ∗ (q)) ⃗ − q⃗ ⋅ m,⃗ Σθζ (m) ⃗ N q∈ℝ

and hence −Σθζ is the Legendre transform of Ξθζ ∘ θ(−Z⃗ ∗ ). By assumption this function is

differentiable at any point in ℝN (equation (7.42)), so Proposition A.10 in Appendix A.2 implies that Σθζ is strictly concave at any interior point of its essential domain, namely, ⃗ N (μ))). ̄ at any m⃗ ∈ Int(Z(Δ ⃗ N (μ)), ̄ then Theorem 6.3 implies that it is an extreme point. This and If m⃗ 0 ∈ 𝜕Z(Δ the strict concavity of Σθζ at inner points imply the uniqueness of m⃗ 0 . 7.2.4 The MinMax theorem: a unified formulation So we finally got our result regarding both existence and uniqueness of a strong generalized (sub)partition verifying the maximal allocation of consumers under given capacities of the agents. The mere existence of optimal strong partition is achieved with little effort. Indeed, Theorem 4.2 implies the existence of weak (sub)partition by “soft” analysis. On the other hand, the proof of Theorem 6.1 implies that for any feasible M⃗ the set of strong w,ζ ⃗ {M}

(sub)partitions is the extreme points of the set 𝒫

(𝒮𝒫

w,ζ ̄ ) ⃗ {M}

of weak ones. Since the

set of extreme points must contain the set of optimal partitions, we get existence of strong partitions in a rather cheap way... The main “hard” analysis we had to follow so far was in order to prove the uniqueness of the optimal partitions, as well as their characterization by the dual problem on 𝕄󸀠 (N, J). One additional bonus we got is that these strong optimal (sub)partitions are open (sub)partitions in the sense of Definition 4.0.1. The duality formalism we extensively used is reflected in the MinMax theorem. The MinMax theorem is of fundamental importance in optimization theory. This theorem, which basically follows from the Hahn–Banach theorem, has many versions. For our case we only need the following, restricted version.

7.2 Optimal strong multipartitions |

101

MinMax theorem. Let 𝕄󸀠 be a vector space over ℝ, 𝒦 being a convex, compact domain. Assume Θ : 𝕄󸀠 × 𝒦 → ℝ is convex in P⃗ ∈ 𝕄󸀠 for any M⃗ ∈ 𝒦 and concave in M⃗ for any P⃗ ∈ 𝕄󸀠 (N, J). Then ⃗ = max inf Θ(P,⃗ M) ⃗ := α. inf max Θ(P,⃗ M)

󸀠 ⃗ ⃗ P∈𝕄 M∈𝒦

󸀠 ⃗ ⃗ P∈𝕄 M∈𝒦

Moreover, there exists M⃗ 0 ∈ 𝒦 such that inf Θ(P,⃗ M⃗ 0 ) = α.

(7.49)

󸀠 ⃗ P∈𝕄

and

In our case we take 𝕄󸀠 = 𝕄󸀠 (N, J), 𝒦 being a convex compact subset of 𝕄+ (N, J), ⃗ := Ξθ,+ (P)⃗ − P⃗ : M⃗ Θ(P,⃗ M) ζ

verifies the conditions of the MinMax theorem. Indeed, we know by now that ⃗ ⃗ ⃗ ̄ (that is, M⃗ is in the essential domain infP∈𝕄 󸀠 (N,J) Θ(P, M) = −∞ unless M ∈ ΔN (μ) ⃗ θ ⃗ ⃗ Since Δ (μ)̄ is a compact subset of 𝕄+ (N, J) we may Θ(P,⃗ M)). of Σ (M) ≡ − inf ⃗ 󸀠 ζ

P∈𝕄 (N,J)

use the MinMax theorem, replacing Θ by

N

⃗ ⃗ Ξθ,+ ζ (P) + H𝒦 (−P), where H𝒦 (P)⃗ := max P⃗ : M⃗

(7.50)

⃗ M∈𝒦

is the support function of 𝒦 (compare with (4.23)) (Appendix A). Using the MinMax theorem, Theorem 7.7 and Proposition 7.1 imply (7.43), which in turn yields max θ(μ)⃗ = max θ(A)⃗ ≡

w,ζ ̄ ⃗ μ∈𝒮𝒫 𝒦

ζ ⃗ A∈𝒮𝒫 𝒦

inf

󸀠 (N,J) ⃗ P∈𝕄

⃗ ⃗ Ξθ,+ ζ (P) + H𝒦 (−P).

(7.51)

In conclusion, we obtain a unified description finding the optimal subpartition ̄ and the oversaturated (𝒦 − ΔN (μ)̄ ≠ 0) cases. for both the undersaturated (𝒦 ⊂ ΔN (μ)) N Likewise, if K ⊂ ℝ , then ⃗∗ ⃗ ⃗ p⃗ 󳨃→ Ξθ,+ ζ (Z (p)) + HK (−p),

(7.52)

HK (p)⃗ := max p⃗ ⋅ m⃗

(7.53)

where ⃗ m∈K

is convex on ℝN .

102 | 7 Optimal multipartitions The MinMax theorem via (7.49) also guarantees the existence of M⃗ 0 ∈ 𝒦 ∩ ΔN (μ)̄ for which (7.51) can be replaced by inf

󸀠 (N,J) ⃗ P∈𝕄

⃗ ⃗ ⃗ Ξθ,+ ζ (P) − P : M0 ,

(7.54)

where the optimal partition is obtained via Theorems 7.8 and 7.9, dealing with the case of a singleton 𝒦 = {M⃗ 0 } (resp. K = {Z(⃗ M⃗ 0 )}). However, the uniqueness of this M⃗ 0 is beyond the mere statement of the MinMax theorem. This uniqueness, and the uniqueness of the corresponding (sub)partition, is the subject of Theorem 7.10. Even if we take for granted the uniqueness of M⃗ 0 , neither the existence nor the uniqueness of a minimizer P⃗ 0 of (7.54) follows from the MinMax theorem. In fact, by Theorem 7.7 we know both existence and uniqueness of this minimizer only if M⃗ 0 hap̄ If M⃗ 0 is a boundary point of ΔN (μ), ̄ then we know pens to be an interior point of ΔN (μ). the uniqueness and existence of an optimal partition by Theorems 6.2 and 6.3, while an equilibrium price vector P⃗ may not exist (Section 7.1.3).

8 Applications to learning theory Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in information? (T. S. Eliot)

8.1 Maximal likelihood of a classifier Let X be the probability space. We can think about it as a space of random samples (e. g., digital data representing figures of different animals). Let ℐ be a finite set of cardinality N. We can think of ℐ as the set of labels, e. g., a lion, elephant, dog, etc. Suppose that Z is a random variable on X × ℐ . We can think about Z as a classifier: For each given data point x ∈ X it produces the random variable x 󳨃→ 𝔼(Z|x) on the set of labels ℐ (see below). Let 𝔼(Z|X) be the X-marginal of Z. We can think of it as a random variable predicting the input data in X. Likewise, 𝔼(Z|ℐ ) is the ℐ -marginal of Z. It can be considered as a random variable predicting the output labels in ℐ . We assume that the input distribution of 𝔼(Z|X) is given by the probability law μ on X. The distribution of Z over X × ℐ is given by a weak partition μ⃗ = (μ1 , . . . , μN ) of (X, μ), where μ = |μ|⃗ := ∑N1 μi . It means that the probability that a data x ∈ X will trigger the label i ∈ ℐ is dμi /dμ(x). Let m⃗ = (m1 , . . . , mN ) ∈ ΔN (1) be the distribution of 𝔼(Z|ℐ ), namely, mi := μi (X) is the probability that Z = i. The Shannon information of 𝔼(Z|ℐ ) is H(Z|ℐ ) = − ∑ mi ln mi . i∈ℐ

It represents the amount of information stored in a random process composed of independent throws of a dice of N = |ℐ | sides, where the probability of getting output i is mi . The Shannon information is always non-negative. Its minimal value H = 0 is attained iff there exists i ∈ ℐ for which mi = 1 (hence mk = 0 for k ≠ i, so the dice falls always on the side i, and we gain no information during the process), and its maximal value is H = ln |ℐ | for a “fair die,” where mi = 1/N. The information corresponds to Z given X is H(Z|X) = − ∑ ∫ ln ( i∈ℐ X

dμi ) dμi . dμ

The marginal information of Z given X is defined by IZ (X, ℐ ) := H(Z|ℐ ) − H(Z|X) = ∑ ∫ ln ( i∈ℐ X

https://doi.org/10.1515/9783110635485-008

dμi ) dμi − ∑ mi ln mi . dμ i∈ℐ

(8.1)

104 | 8 Applications to learning theory This information is always non-negative via Jensen’s inequality and the convexity of −H as a function of the distribution. This agrees with the interpretation that the correlation between the signal X and the output label ℐ contributes to the marginal information. In particular, if the marginals 𝔼(Z|ℐ ) and 𝔼(Z|X) are independent (so μi = mi μ), then H(Z|X) = H(Z|ℐ ), so IZ (X, ℐ ) = 0. Let θ(i, x) := θi (x) measure the level of likelihood that an input data x corresponds to a label i. The average likelihood due to a classifier Z is, thus, μ(⃗ θ)⃗ := ∑ μi (θi ).

(8.2)

i∈ℐ

The object of a learning machine is to develop a classifier Z which will produce a maximal likelihood under a controlled amount of marginal information. In the worst case scenario, a relevant function is the minimal possible marginal information for a given likelihood μ(⃗ θ)⃗ = α. For this we define this minimal information as R(α) := inf{IZ (X, ℐ ), μ(⃗ θ)⃗ = α, 𝔼(Z|ℐ ) = m,⃗ 𝔼(Z|X) = μ}. Z

From (8.1), (8.2) we may rewrite } { dμ ⃗ = m,⃗ |μ|⃗ = μ} − ∑ mi ln mi . R(α) = inf { ∑ ∫ ln ( i ) dμi ; θ(μ)⃗ = α, μ(X) dμ μ⃗ } i∈ℐ {i∈ℐ X ⃗ From this definition and the linearity of μ⃗ → μ(θ) it follows that R is a concave function. The concave dual of R is R∗ (β) := inf R(α) − αβ, α

which is a concave function as well. By the MinMax theorem we recover R(α) = sup R∗ (β) + αβ. β

Proposition 8.1. Let Q(β, ϕ) := β ∑ mi ln (∫ e i∈ℐ

θi +ϕ β

dμ) − ∫ ϕdμ.

X

(8.3)

X

Then R∗ (β) := inf Q(β, ϕ). ϕ∈C(X)

(8.4)

8.1 Maximal likelihood of a classifier

| 105

Note that Q(β, ϕ) = Q(β, ϕ + λ) for any constant λ ∈ ℝ. Thus, we use (8.3), (8.4) to write R∗ (β) =

inf

ϕ∈C(X);∫X ϕdμ=0

β ∑ mi ln (∫ e i∈ℐ

θi +ϕ β

dμ) .

(8.5)

X

The parameter β can be considered as the “temperature,” which indicates the amount of uncertainty of the optimal classifier Z. In the “freezing limit” β → 0 we get lim β ln (∫ e

β→0

dμ) = max θi (x) + ϕ(x), x∈X

X

so R∗ (0) =

θi +ϕ β

inf {∑ mi max[θi (x) + ϕ(x)]} .

∫ ϕdμ=0

i∈ℐ

x∈X

(8.6)

It can be shown that R∗ (0) is obtained by the optimal partition of X corresponding to the utility {θi } and capacities m⃗ which we encountered in Chapter 4: } { } { ⃗ := sup { ∑ ∫ θi dμ; μ(Ai ) = mi } , R∗ (0) = Σθ (m) } { A⃗ i∈ℐ A i } {

(8.7)

where A⃗ = {Ai , . . . , AN } is a strong partition of X. Indeed, a minimizing sequence of ϕ in (8.6) converges pointwise to a limit which is a constant −pi on each of the optimal components Ai in A,⃗ and these constants are the equilibrium prices which minimize Ξ(p)⃗ − p⃗ ⋅ m⃗ on ℝN , where Ξ(p)⃗ := ∫ max[θi − pi ]dμ. X

i∈ℐ

Compare with (4.9), (4.10). In particular, the optimal classifier at the freezing state corresponds to μi = μ⌊Ai , where Ai ⊂ {x ∈ X; θi (x) − pi = max θj (x) − pj }, j∈ℐ

and verifies the conditions of strong partitions Ai ∩ Aj = 0 for i ≠ j and ⋃i Ai = X. Thus, the optimal Z will predict the output i for a data x ∈ X with probability 1 iff x ∈ Ai , and with probability 0 if x ∈ ̸ Ai . In the limit β = ∞ we look for a classifier Z satisfying IZ (X, ℐ ) = 0, that is, the amount of information H(Z|X) is the maximal one. Since lim β ln (∫ e

β→∞

X

θi +ϕ β

dμ) = ∫(θi + ϕ)dμ = ∫ θi dμ, X

it follows that the optimal likelihood in the limit β = ∞ is

X

106 | 8 Applications to learning theory

R∗ (∞) = ∑ mi μ(θi ), i∈ℐ

⃗ corresponding to the independent variables 𝔼(Z|X), 𝔼(Z|ℐ ), where μ⃗ = mμ. Proof of Proposition 8.1. Let us maximize ∑ ∫ θi dμi − β ∑ ∫ ln (

i∈ℐ X

i∈ℐ X

dμi ) dμi + ∑ ∫ ϕdμi − ∫ ϕdμ dμ i∈ℐ X

(8.8)

X

under the constraints μi (X) = mi . Here θ ∈ C(X) is the Lagrange multiplier for the constraint |μ|⃗ = μ. Taking the variation of (8.8) with respect to μi we get θi − β ln (

dμi ) + ϕ = γi , dμ

where γi is the Lagrange multiplier due to the constraint μi (X) = mi . Thus θi +ϕ

dμi mi e β . = θi +ϕ dμ ∫X e β dμ Substitute this in (8.8) to obtain that (8.8) is maximized at Q(β, ϕ) − β ∑i∈ℐ mi ln mi . Minimizing over ϕ ∈ C(X) we obtain (8.4).

8.2 Information bottleneck The information bottleneck (IB) method was first introduced by Tishby, Pereira, and Bialek [46] in 1999. Here we attempt to obtain a geometric characterization of this concept. Suppose a classifier U is given on X × 𝒥 , where the label set 𝒥 is finite of cardinality |𝒥 | < ∞. The object of a learning machine is to reduce the details of the data space X to a finite space ℐ whose cardinality is |ℐ | ≤ |𝒥 |. Such a learning machine can be described by a random variable (r. v.) V on X × ℐ which is faithful, i. e., the X-marginal of V on X coincides with that of U: 𝔼(U|X) = 𝔼(V|X).

(8.9)

We denote this common distribution on X by μ. Such a random variable will provide a classifier W on ℐ × 𝒥 by composition: Prob(i ∈ ℐ , j ∈ 𝒥 |W) := 𝔼(U = (x, i), V = (x, j)|ℐ , 𝒥 ). We note on passing that such a composition never increases the marginal information, so IU (X, 𝒥 ) ≥ IW (ℐ , 𝒥 ) (see below).

(8.10)

8.2 Information bottleneck | 107

As in Section 8.1 we represent the given distribution of U in terms of a weak J-partition μ̄ := (μ(1) , . . . , μ(j) ), where μ(j) is a positive measure on X and |μ|̄ := ∑j∈𝒥 μ(j) = μ is the marginal distribution of U on X. Let us denote ζj :=

dμ(j) ; dμ

ζ ̄ := (ζ1 , . . . , ζJ ),

so μ(j) = ζj μ and |ζ ̄ | := ∑j∈𝒥 ζj = 1 on X (compare with (5.1)). The (unknown) distribution of the classifier V can be introduced in terms of a weak N-partition of μ: μ⃗ = (μ1 , . . . , μN ), where |μ|⃗ := ∑i∈ℐ μi = μ, via (8.9). The decomposition of U and V provides the classifier W on ℐ × 𝒥 . The distribution (j) of this classifier is given by an N × J matrix M⃗ = {mi }, where mi := μi (ζj )

(8.11)

(j)

is the probability that W = (i, j). Let (j) (j) m⃗ (j) := (m1 , . . . , mI ),

|m⃗ (j) | := m(j) = μ(j) (X), (J) m̄ i := (m(1) i , . . . , mi ),

|m̄ i | := mi = μi (X).

(8.12)

The information of W and its ℐ - and 𝒥 -marginals are given by H(W|ℐ , 𝒥 ) = − ∑ ∑ mi ln (mi ) , (j)

(j)

i∈ℐ j∈𝒥

H(W|ℐ ) + H(W|𝒥 ) = − ∑ m(j) ln m(j) − ∑ mi ln mi . j∈𝒥

i∈ℐ

The marginal information of W is given as IW (ℐ , 𝒥 ) = H(W|ℐ ) + H(W|𝒥 ) − H(W|ℐ , 𝒥 ) = − ∑ mi ln mi − ∑ m(j) ln m(j) + ∑ ∑ mi ln mi . (j)

i∈ℐ

j∈𝒥

(j)

i∈ℐ j∈𝒥

(8.13)

Note that H(U|X) = − ∑ ∫ ζj ln (ζj ) dμ, H(U|𝒥 ) = H(W|𝒥 ) = − ∑ m(j) ln m(j) , j∈𝒥 X

j∈𝒥

so the marginal information due to U is IU (X, 𝒥 ) = − ∑ m(j) ln m(j) + ∑ ∫ ζj ln (ζj ) dμ. j∈𝒥

j∈𝒥 X

(8.14)

108 | 8 Applications to learning theory ⃗ and mi := ∫ ζj dμi ⃗ Note that s 󳨃→ s ln s is a convex function. Since |μ|⃗ = μ, |μ(X)| := |m|, we get by Jensen’s inequality (j)

∫ ζj ln (ζj ) dμ = ∑ mi ∫ ζj ln (ζj ) i∈ℐ

X

X

m dμi (j) ≥ ∑ mi ln i , mi i∈ℐ mi (j)

so (recalling ∑j∈𝒥 mi = mi ) (j)

∑ ∫ ζj ln (ζj ) dμ ≥ ∑ ∑ mi ln ( (j)

j∈𝒥 X

i∈ℐ j∈𝒥

mi

(j)

mi

),

so by (8.13), (8.14) we verify (8.10). Recall that the Jensen inequality turns into an (j) equality iff ζj = mi /mi μi -a. e. Thus we have the following. The difference between the marginal information in W and the marginal information in U is the distortion (j) m (j) IU (𝒥 , X ) − IW (ℐ, 𝒥 ) = ∑ ∫ ζj ln (ζj ) dμ − ∑ ∑ mi ln ( i ) ≥ 0. mi j∈𝒥 i∈ℐ j∈𝒥 X

The information gap can be made zero only if |ℐ| ≥ |𝒥 | and the classifier U is a deterministic one, i. e., ζj ∈ {0, 1}. In that case ℐ ⊃ 𝒥 and any choice of an immersion τ : 𝒥 → ℐ implies that V = τ ∘ U is an optimal choice to minimize the information gap.

Figure 8.1: A diagram of the random variables vs the spaces.

8.2.1 Minimizing the distortion For a given r. v. U subjected to the distribution μ,̄ all possible distributions of W for a ̄ In particular, we can look for given cardinality |ℐ | are represented by points in ΔN (μ). the optimal W which minimizes the information gap with respect to U in terms of its ̄ Since m(j) are independent of U, it follows by (8.13) that M⃗ 0 distribution M⃗ 0 ∈ ΔN (μ). is a maximizer of ⃗ := ∑ ∑ m ln ( h(M) i (j)

i∈ℐ j∈𝒥

mi

(j)

mi

̄ ) ∀M⃗ ∈ ΔN (μ),

8.2 Information bottleneck | 109

where we recall (8.12). Thus ⃗ = ∑ h(m̄ i ), h(M)

̄ := ∑ m(j) ln m(j) − |m|̄ ln(|m|). ̄ h(m)

i∈ℐ

j∈𝒥

(8.15)

Lemma 8.1. h is positively homogeneous (Definition A.6.2) and strongly convex on the ̄ simplex ΔN (μ). ̄ = λh(m) ̄ for any λ ≥ 0, m̄ ∈ ℝJ . Proof. Direct observation implies h(λm) Differentiating h twice in ℝJ++ we obtain −1

−1 𝜕h j = δi (m(i) ) − ( ∑ m(k) ) . 𝜕m(i) 𝜕m(j) k∈𝒥

Given a vector ᾱ = (α1 , . . . , αJ ) ∈ ℝJ we obtain ∑

i,j∈𝒥

αj2 (∑j∈𝒥 αj )2 𝜕h α α = − . ∑ i j (j) 𝜕m(i) 𝜕m(j) ∑j∈𝒥 m(j) j∈𝒥 m

(8.16)

Using the Cauchy–Schwartz inequality, ∑ αj = ∑

j∈𝒥

j∈𝒥

αj √m(j)

√m(j) ≤ ( ∑

j∈𝒥

αj2

m

1/2

) (j)

1/2

( ∑ m(j) ) j∈𝒥

,

which implies that (8.16) is non-negative. Moreover, an equality in Cauchy–Schwartz implies ᾱ = λm̄ for some λ > 0. It follows that h is strongly convex on the simplex ̄ it follows that h is strongly convex ΔJ (1). Since ΔN (μ)̄ is constrained by ∑i∈ℐ m̄ i = μ(X) ̄ in ΔN (μ). From the convexity of h and (8.15) we obtain the convexity of IW (ℐ , 𝒥 ) as a function ̄ Since ΔN (μ)̄ is a compact and convex set we obtain immediately the of M⃗ on ΔN (μ). (j) ̄ Moreover, the existence of a maximizer M⃗ 0 := {mi,0 } in the relative boundary of ΔN (μ). ̄ set of maximizers is a convex subset of ΔN (μ). Lemma 8.2. The maximizer M⃗ 0 of h in ΔN (μ)̄ is unique iff the vectors (m̄ i,0 , h(m̄ i,0 )) ∈ ℝJ+1 , i = 1 . . . N, are independent in ℝJ+1 . In particular, N ≤ J + 1 is a necessary condition for uniqueness of the maximizer of IW (ℐ, 𝒥 ).

⃗ = h(M⃗ 0 ) iff there Proof. By the strong convexity of h (Lemma 8.1) we obtain that h(M) ⃗ = exist λ1 , . . . , λN > 0 such that m̄ i = λi m̄ i,0 for i = 1, . . . , N. If this is the case, then h(M) ∑i∈ℐ λi h(m̄ i,0 ). Thus ∑ λi h(m̄ i,0 ) = h(M⃗ 0 ).

i∈ℐ

(8.17)

110 | 8 Applications to learning theory In addition, we recall from (8.12) that any M⃗ ∈ ΔN (μ)̄ is subjected to the constraint ̄ ̄ which, together with (8.17), implies Hence ∑i∈ℐ λi m̄ i,0 , = μ(X), ∑i∈ℐ m̄ i = μ(X). ̄ h(M⃗ 0 )) . ∑ λi (m̄ i,0 , h(m̄ i,0 )) = (μ(X),

i∈ℐ

(8.18)

The system (8.17), (8.18) admits the solution λ1 = ⋅ ⋅ ⋅ = λN = 1, and this is the unique solution of this system iff the vectors (m̄ i,0 , h(m̄ i,0 )), i ∈ ℐ , are independent in ℝJ+1 . Remark 8.2.1. The uniqueness of the maximizer M⃗ 0 does not necessarily imply the uniqueness of the optimal classifier V realizing the minimal information gap. In fact, a classifier V is determined by the partition μ⃗ = (μ1 , . . . , μN ) of X, and the uniqueness of M⃗ 0 only implies that the corresponding partition must satisfy μi (X) = |m̄ i,0 | := (j) ∑j∈𝒥 mi,0 . ̄ Recall that M⃗ 0 is a boundary point of ΔN (μ). ⃗ in ΔN (μ)̄ and satisfy the conTheorem 8.1. Let M⃗ 0 ∈ 𝜕ΔN (μ)̄ be a maximizer of h(M) dition of Lemma 8.2. Assume ζ ̄ satisfies Assumption 6.2.1. Let z⃗(i) ∈ ℝJ , i = 1, . . . , N, ⃗ N (μ)) ̄ (Definition 6.2.2), then the be given satisfying Assumption 6.2.2. If Z(⃗ M⃗ 0 ) ∈ 𝜕Z(Δ minimal information gap for a given cardinality N is a unique, deterministic classifier V. In particular, its distribution is given by a partition μ⃗ = μ⌊A,⃗ that is, μi = μ⌊Ai , where A⃗ = (A1 , . . . , AN ) is a strong partition. ⃗ ̄ ⊂ The proof of this theorem follows from Theorem 6.3 (ii). Note that 𝜕Z(Δ N (μ)) ⃗ ⃗ ̄ (since ΔN (μ)̄ ⊂ ΔN (μ)) ̄ and any subpartition corresponding to M ∈ ΔN (μ)̄ is 𝜕Z(ΔN (μ)) necessarily a partition.

8.2.2 The information bottleneck in the dual space We are given a random variable U on the state space X × 𝒥 as an input. As before, we can view U as a classifier over the set of features X into the set of labels 𝒥 . A network gets this classifier as an input, and (stochastically) represents the data x ∈ X by internal states i ∈ ℐ of the network. We assume that ℐ is a finite set |ℐ | = N. As a result of the training we get a classifier on the set ℐ × 𝒥 , where ℐ is the reduction of the feature space X. The objectives of the “information bottleneck” as described by Tishbi and coauthors are: – Predictability: to preserve as much of the marginal information of the induced classifier W as possible, that is, to minimize the information gap between U and W. – Compressibility: to minimize as much as possible the marginal information stored in the classifier V.

8.2 Information bottleneck | 111

In addition we include the possibility of a likelihood function θ : ℐ × X → ℝ as in Section 8.1. So, we add another objective: – To increase as much as possible the expected likelihood of V as a classifier. Now, we consider the information bottleneck (IB) variational problem. The IB was originally introduced by Tishby and coauthors [46], who suggested to minimize1 P(V , β, γ) := IV (ℐ, X ) − βIW (ℐ, 𝒥 ) − γ𝔼(θ(V )),

(IB)

where β, γ ≥ 0 (in the current literature γ = 0). The rationale behind IB is as follows: The desired classifier V should induce maximal marginal information on the induced W, as well as maximal likelihood. On the other hand, the price paid for maximizing this information is the complexity of V measured in terms of the marginal information stored (IV ). The limit of β large corresponds to maximal information in W (i. e., the minimal information gap). Likewise, the limit of large γ emphasizes the importance of the likelihood of V. Let us calculate the marginal information IV (ℐ , X): H(V|X) = − ∑ ∫ ln ( i∈ℐ X

dμi ) dμi , dμ

while H(V|ℐ ) = H(W|ℐ ) = − ∑ mi ln mi , i∈ℐ

so IV (ℐ , X) = − ∑ mi ln mi + ∑ ∫ ln ( i∈ℐ

i∈ℐ X

dμi ) dμi . dμ

(8.19)

Finally, we recall that the expected likelihood of V is ⃗ 𝔼(θ(V)) := ∑ ∫ θi dμi := μ(⃗ θ). i∈ℐ X

Note that m(j) are independent of V. In terms of the distribution μ⃗ of V we obtain ⃗ where P(V) − β ∑j∈𝒥 m(j) ln m(j) ≡ P(μ), P(μ)⃗ = ∑ ∫ (ln ( i∈ℐ X

m dμi (j) ) − γθi ) dμi − ∑ mi ln mi − β ∑ ∑ mi ln ( i ) . dμ mi i∈ℐ i∈ℐ j∈𝒥

1 Compare with [1, 19], where β corresponds to β−1 .

(j)

(8.20)

112 | 8 Applications to learning theory Let P1 (μ)⃗ := ∑ ∫ (ln ( i∈ℐ X

dμi ) − γθi ) dμi . dμ

Here 𝕄󸀠 (N, J) is as given in Definition 5.2.1, P⃗ := (p⃗ 1 , . . . , p⃗ N ). Lemma 8.3. We have inf P1 (μ)⃗ =

w,ζ ⃗ μ∈𝒫 ⃗

{M}

w,ζ ⃗ {M}

Proof. Recall that 𝒫 that ∫X ζj dμi =

(j) mi

N { } ⃗ ̄ ln ( e−pk ⋅ζ +γθk ) dμ + P⃗ : M⃗ } + 1. ∑ ∫ { 󸀠 ⃗ P∈𝕄 (N,J) k=1 {X }

inf

≠ 0 iff there exists a weak partition μ⃗ = (μ1 , . . . , μN ) of μ such

and |μ|⃗ = μ. In particular,

{ 0 sup ∑ ∫ (ϕ − p⃗ i ⋅ ζ ̄ ) dμi + P⃗ : M⃗ − ∫ ϕdμ = { ∞ 󸀠 (N,J),ϕ∈C(X) ⃗ P∈𝕄 i∈ℐ X X {

w,ζ , ⃗ {M} w,ζ ∉ 𝒫 ⃗ , {M}

if μ⃗ ∈ 𝒫 if μ⃗

where p⃗ i ∈ ℝJ , P⃗ = (p⃗ 1 , . . . , p⃗ N ) ∈ 𝕄󸀠 (N, J), and ϕ ∈ C(X). Then sup

∑ ∫ (ln (

󸀠 (N,J),ϕ∈C(X) ⃗ P∈𝕄 i∈ℐ

X

dμi ) − γθi + ϕ − p⃗ i ⋅ ζ ̄ ) dμi + P⃗ : M⃗ − ∫ ϕdμ dμ X

{ P1 (μ)⃗ ={ ∞ {

It follows that inf P1 (μ)⃗ = inf sup ∑ ∫ (ln ( μ⃗

w,ζ ⃗ μ∈𝒫 ⃗ {M}

⃗ P,ϕ i∈ℐ X

w,ζ , ⃗ {M} w,ζ ∉ 𝒫 ⃗ . {M}

if μ⃗ ∈ 𝒫 if μ⃗

(8.21)

dμi ) − γθi + ϕ − p⃗ i ⋅ ζ ̄ ) dμi + P⃗ : M⃗ − ∫ ϕdμ, dμ X

where the supremum is over P⃗ ∈ 𝕄󸀠 (N, J), ϕ ∈ C(X) and the infimum is unconstrained. By the MinMax theorem inf P1 (μ)⃗ = sup inf ∑ ∫ (ln (

w,ζ ⃗ μ∈𝒫 ⃗ {M}

⃗ P,ϕ

μ⃗

i∈ℐ X

dμi ) − γθi + ϕ − p⃗ i ⋅ ζ ̄ ) dμi + P⃗ : M⃗ − ∫ ϕdμ, dμ

(8.22)

X

and, moreover, inf P1 (μ)⃗ < ∞,

w,ζ ⃗ μ∈𝒫 ⃗

(8.23)

{M}

w,ζ ⃗ {M}

since 𝒫

≠ 0. We now consider the unconstrained infimum dμ Q1 (ϕ, P)⃗ := inf ∑ ∫ (ln ( i ) − γθi + ϕ + p⃗ i ⋅ ζ ̄ ) dμi . dμ μ⃗ i∈ℐ X

(8.24)

8.2 Information bottleneck | 113

We find that the minimizer of (8.24) exists, and takes the form dμi ⃗ ̄ = eγθi −ϕ+pi ⋅ζ −1 . dμ

(8.25)

The condition |μ|⃗ = μ implies that ϕ + 1 = ln ( ∑ eγθk +pk ⋅ζ ) ⃗

̄

k∈ℐ

(8.26)

and from (8.22) N } { ⃗ ̄ inf P1 (μ)⃗ = sup {− ∫ ln ( ∑ epk ⋅ζ +γθk ) dμ + P⃗ : M⃗ } − 1 w,ζ ⃗ μ∈𝒫 P⃗ k=1 ⃗ {M} } { X N { } ⃗ ̄ = − inf {∫ ln ( ∑ epk ⋅ζ +γθk ) dμ − P⃗ : M⃗ } − 1. P⃗ k=1 {X }

(8.27)

Lemma 8.4. We have N { } ⃗ ̄ inf {∫ ln ( ∑ epk ⋅ζ +γθk ) dμ − P⃗ : M⃗ } > −∞ ⃗ P k=1 {X }

̄ where ΔN (μ)̄ is as defined in (5.1.2). iff M⃗ ∈ ΔN (μ), Proof. Recall from Theorem 5.1 that M⃗ ∈ ̸ ΔN (μ)̄ iff inf

󸀠 (N,J) ⃗ P∈𝕄

Ξ0ζ (P)⃗ − P⃗ : M⃗ = −∞,

(8.28)

where Ξ0ζ (P)⃗ = ∫X maxi p⃗ i ⋅ ζ ̄ dμ.

Since ln(∑Nk=1 ep⃗ k ⋅ζ +γθk ) = maxi p⃗ i ⋅ ζ ̄ + O(1), (8.28) implies the bound. ̄

Theorem 8.2. The minimal value of IB is the minimum of (j) N { } m ⃗ ̄ (j) − inf {∫ ln ( ∑ epk ⋅ζ +γθk ) dμ − P⃗ : M⃗ } − ∑ mi ln mi − β ∑ ∑ mi ln ( i ) − 1 mi P⃗ i∈ℐ j∈𝒥 k=1 {X } i∈ℐ (8.29)

̄ If the infimum in P⃗ 0 ∈ 𝕄󸀠 (N, J) is attained for a minimizer M⃗ 0 ∈ over M⃗ ∈ ΔN (μ). 𝕄+ (N, J), then the distribution of the minimizer V of IB is given by the weak partition 0

μi (dx) =

eγθi (x)+p⃗ i ⋅ζ (x) ̄

0

∑k eγθk (x)+p⃗ k ⋅ζ (x) ̄

μ(dx).

(8.30)

114 | 8 Applications to learning theory Proof. It follows from (8.20), Lemma 8.3, and Lemma 8.4. The minimizer μ⃗ follows from (8.25), (8.26). In the notation of [46], where γ = 0, the optimal distribution μi takes the form dμi = 𝒵 −1 Mi e−βDKL , dμ

(8.31)

where DKL (U|W) is the Kullback–Leibler divergence [14] for the distribution of (U, W) and 𝒵 is the partition function which verifies the constraint ∑i∈ℐ μi = μ. In our notation DKL (U|W) = ∑ ζj (x) ln (

ζj (x)mi

j∈𝒥

mi

(j)

).

To relate (8.30) with (8.31) we assume that the optimal M⃗ in (8.29) is a relative internal ̄ Then we equate the derivative of (8.29) with respect to m(j) point of ΔN (μ). to zero, at i the optimal P⃗ 0 , to obtain pi

(j),o

= ln (mi ) + β ln (

mi

(j)

mi

) + λ(j) ,

where λ(j) is the Lagrange multiplier corresponding to the constraint ∑i∈ℐ mi = m(j) . (j)

Since ∑j∈𝒥 ζ (j) (x) = 1 we get p⃗ 0i ⋅ ζ ̄ = ln mi + β ∑j∈𝒥 ζj (x) ln( takes the form (where γ = 0) μi (dx) =

mi 𝒵

e

β ∑j∈𝒥 ζj (x) ln(

(j) i mi

m

)+λ⋅̄ ζ ̄ (x)

m(j) i ) mi

+ λ̄ ⋅ ζ ̄ (x). Thus, (8.30)

μ(dx),

where 𝒵 is the corresponding partition function. Now, we can add and subtract any function of x to the powers of the exponents since any such function is canceled out with the updated definition of 𝒵 . We add the function x 󳨃→ ∑j∈𝒥 ζj ln ζj and subtract λ̄ ⋅ ζ ̄ to get (8.31). ̄ The representation (8.31) is valid only if the minimizer of (8.29) is a relative interior point of ΔN (μ). From Section 8.2.1 we realize that this may not be the case if β is sufficiently large.

|

Part III: From optimal partition to optimal transport and back

9 Optimal transport for scalar measures A plan is the transport medium which conveys a person from the station of dreams to the destination of success. Goals are the transport fees. (Israelmore Ayivor)

9.1 General setting So far we considered the transport problem from the source, given by a measure space ⃗ Here we consider the ex(X, μ), to a target, given by a discrete measure space (ℐ , m). tension where the target is a general measure space (Y, ν). We pose the following assumption. Assumption 9.1.1. X, Y are compact spaces, θ ∈ C(X, Y) is non-negative, and μ ∈ ℳ+ (X), ν ∈ ℳ+ (Y) are regular Borel measures. We define θ(μ, ν) := max ∫ ∫ θ(x, y)π(dxdy), π∈Π(μ,ν)

(9.1)

X Y

where Π(μ, ν) := {π ∈ ℳ+ (X × Y); μ(dx) ≥ π(dx, Y), ν(dy) ≥ π(X, dy)} .

(9.2)

In the balanced case μ(X) = ν(Y) we may replace Π by Π(μ, ν) := {π ∈ ℳ+ (X × Y) ; μ(dx) = π(dx, Y), ν(dy) = π(X, dy)} .

(9.3)

The optimal π is called an optimal transport plan (OTP). Example 9.1.1. If μ = αδx , ν = βδy , where α, β > 0, then Π(μ, ν) := {(α ∧ β)δx δy } is a single measure and θ(αδx , βδy ) = (α ∧ β)θ(x, y). Example 9.1.2. Assume μ ∈ ℳ+ (X), ν = ∑i∈ℐ mi δyi , μ(X) ≥ ∑i∈ℐ mi , and Π(μ, ν) = {∑ δyi (dy) ⊗ μi (dx), where ∫ μi = mi and ∑ μi ≤ μ}. i∈ℐ

X

i∈ℐ

In that case θ(μ, ν) corresponds to the undersaturated case (4.7). If μ(X) ≤ ∑i∈ℐ mi , then Π(μ, ν) = {∑ δyi (dy) ⊗ μi (dx), where ∫ μi ≤ mi and ∑ μi = μ}, i∈ℐ

X

then θ(μ, ν) corresponds to the oversaturated case (4.5). https://doi.org/10.1515/9783110635485-009

i∈ℐ

118 | 9 Optimal transport for scalar measures As we see from Example 9.1.2, these definitions also extend our definition of weak partitions in Chapter 5, where Y := ℐ := {1, . . . , N} and ν := ∑i∈ℐ mi δ(i) .

9.2 Duality Recall that in Chapter 4 (equations (4.9), (4.10)), we considered strong (sub)partition, where the maximizers of (9.1), (1.16) are obtained as the deterministic partition μi = μ⌊Ai . The analog of strong (sub)partitions in the general transport case is an optimal transport map (OTM) T : X → Y such that, formally, the optimal plan π takes the form πT (dxdy) = μ(dx)δy−T(x) dy. Thus: i) If μ(X) < ν(Y), then πT ∈ Π(μ, ν) iff T# μ ≤ ν, that is, for any Borel set B ⊂ Y μ(T −1 (B)) ≤ ν(B). Equivalently, ∫ ϕ(T(x))dμ(x) ≤ ∫ ϕ(y)dν(y) ∀ϕ ∈ C(X). X

Y

ii) If μ(X) > ν(Y), then πT ∈ Π(μ, ν) iff T# μ ≥ ν, that is, for any Borel set B ⊂ Y μ(T −1 (B)) ≥ ν(B). Equivalently, ∫ ϕ(T(x))dμ(x) ≥ ∫ ϕ(y)dν(y) ∀ϕ ∈ C(X). X

Y

iii) If μ(X) = ν(Y), then Π(μ, ν) = Π(μ, ν) and πT ∈ Π(μ, ν) iff T# μ = ν, that is, for any Borel set B ⊂ Y μ(T −1 (B)) = ν(B). Equivalently, ∫ ϕ(T(x))dμ(x) = ∫ ϕ(y)dν(y) ∀ϕ ∈ C(X). X

Y

The way from the “stochastic” OTP to the deterministic OTM which we did for the semi-discrete case is concealed in the dual formulation. If the target space Y is a finite space, then we obtained, under Assumption 7.2.1 (in case J = 1), that the optimal weak (sub)partition is given by the strong (sub)partition determined by the prices p ∈ ℝ|Y| . To show the connection with Monge–Kantorovich theory ([48, 49]), define 𝒥 θ := {(ξ , p) ∈ C(X) × C(Y); ξ (x) + p(y) ≥ θ(x, y) ∀(x, y) ∈ X × Y}.

(9.4)

Consider first the saturation case μ(X) = ν(Y). Then, for any π ∈ Π(μ, ν) and any (ξ , p) ∈ 𝒥 θ , ∫ θdπ ≤ ∫ [ξ (x) + p(y)]π(dxdy) = ∫ ξdμ + ∫ pdν, X×Y

X×Y

X

Y

(9.5)

9.2 Duality | 119

and hence, in particular, θ(μ, ν) ≤

inf

(ξ ,p)∈𝒥 θ

∫ ξdμ + ∫ pdν. X

(9.6)

Y

Assume μ(X) > ν(Y). Then (9.6) cannot be valid since the infimum on the right is −∞. Indeed, we obtain for any constant λ that (ξ , p) ∈ 𝒥 θ iff (ξ − λ, p + λ) ∈ 𝒥 θ , and ∫(ξ − λ)dμ + ∫(p + λ)dν = ∫ ξdμ + ∫ pdν + λ(ν(Y) − μ(X)) → −∞ X

Y

X

Y

as λ → ∞. However, (9.6) is still valid for π ∈ Π(μ, ν) if we restrict the pair (ξ , p) to (ξ , p) ∈ 𝒥 θ such that ξ ≥ 0. Indeed, (9.5) implies that ∫ θdπ ≤ ∫ ξdμ̂ + ∫ pdν X×Y

X

Y

̂ ̂ for any π ∈ Π(μ, ν), where μ(dx) = π(dx, Y) ≤ μ satisfying μ(X) = ν(Y). If ξ ≥ 0, then ∫X ξdμ ≥ ∫X ξdμ,̂ thus θ(μ, ν) ≤

inf

(ξ ,p)∈𝒥 θ ,ξ ≥0

∫ ξdμ + ∫ pdν X

(9.7)

Y

holds in the case μ(X) > ν(Y) for any π ∈ Π(μ, ν). ̂ Now, suppose (9.7) is satisfied with equality. Let ν̂ > ν such that ν(Y) < μ(X). Since θ ≥ 0 by assumption, θ(μ, ν)̂ ≥ θ(μ, ν) by definition. If (ξϵ , pϵ ) ∈ 𝒥 θ , ξϵ ≥ 0 satisfies ∫X ξϵ dμ + ∫Y pϵ dν ≤ θ(μ, ν) + ϵ for some ϵ > 0, then from ∫X ξϵ dμ + ∫Y pϵ dν̂ ≥ θ(μ, ν)̂ we obtain ∫ pϵ (dν̂ − dν) ≥ −ϵ. Y

̂ Since we may take ν(Y) as close as we wish to μ(X) (e. g., ν̃ = ν + αδy0 for any α < μ(X) − ν(Y) and any y0 ∈ Y), we get pϵ ≥ −ϵ/(μ(X) − ν(Y)). Since ϵ > 0 is arbitrary we obtain that the infimum must be attained at p ≥ 0. In particular (compare with Proposition 4.6 and the remark thereafter), we have the following. Proposition 9.1. Suppose θ ≥ 0, μ(X) > ν(Y), and (9.7) is an equality. Then θ(μ, ν) =

inf

(ξ ,p)∈𝒥 θ ,ξ ≥0,p≥0

∫ ξdμ + ∫ pdν. X

(9.8)

Y

By the same reasoning (flipping μ with ν) we obtain the following. If μ(X) < ν(Y) and θ(μ, ν) = inf(ξ ,p)∈𝒥 ,p≥0 ∫X ξdμ + ∫Y pdν, then (9.8) holds as well. θ

120 | 9 Optimal transport for scalar measures It is remarkable that in the case of saturation μ(X) = ν(Y), an equality in (9.7) does not, in general, imply (9.8). Evidently, we may restrict 𝒥 θ to either p ≥ 0 or ξ ≥ 0 by replacing (ξ , p) with (ξ + λ, p − λ) for an appropriate constant λ, but not both! To remove the conditioning in Proposition 9.1 we use the corresponding equalities in the saturation case. This is the celebrated duality theorem discovered by Kantorovich [28] and Koopmans [29] – for which they shared the Nobel Prize in economics. Theorem 9.1. If μ(X) = ν(Y), θ(μ, ν) =

inf

(ξ ,p)∈𝒥 θ

∫ ξdμ + ∫ pdν. X

Y

Remark 9.2.1. In the balanced case we can surely remove the assumption that θ is non-negative. Indeed, we may always change θ by an additive constant. However, in the imbalanced case μ(X) ≠ ν(Y), we cannot remove the assumption θ ≥ 0. If, e. g., θ is a non-positive function, then θ(μ, ν) = 0 by choosing π = 0 in Π(μ, ν) (equation (9.2)). We extend this theorem to the unbalanced cases as follows. Theorem 9.2. Suppose μ(X) ≠ ν(Y). Then θ(μ, ν) =

inf

(ξ ,p)∈𝒥 θ ,ξ ≥0,p≥0

∫ ξdμ + ∫ pdν X

Y

holds. Proof. We prove the first claim for θ in the case μ(X) > ν(Y). The other claims follow by symmetry. By definition and the assumption θ ≥ 0 we obtain θ(μ, ν) =

sup

̃ ̃ μ≤μ, μ(X)=ν(Y)

θ(μ,̃ ν).

By Theorem 9.1 θ(μ, ν) =

{ } ∫ ξdμ̃ + ∫ pdν} . { ̃ ̃ (ξ ,p)∈𝒥 θ μ≤μ, μ(X)=ν(Y) Y {X } sup

inf

̃ Since X is compact, the set μ̃ ≤ μ, μ(X) = ν(Y) is compact in ℳ+ (X) with respect to the C ∗ (X) topology. Hence, the MinMax theorem implies θ(μ, ν) =

{ } ̃ + ∫ pdν} . ∫ ξdμ) {( ̃ sup ̃ (ξ ,p)∈𝒥 θ μ≤μ, μ(X)=ν(Y) X Y { } inf

(9.9)

For a given ξ ∈ C(X) let Ā ξ (λ) := {x ∈ X; ξ (x) ≥ λ} and Aξ (λ) := {x ∈ X; ξ (x) > λ}. The function λ 󳨃→ μ(Ā ξ (λ)) is monotone non-increasing, lower semi-continuous, while λ 󳨃→

9.3 Deterministic transport | 121

μ(Aξ (λ)) is monotone non-increasing, upper semi-continuous. In addition, μ(Aξ (λ)) ≤ μ(Ā ξ (λ)) for any λ. Thus, there exists λ0 such that μ(Ā ξ (λ0 )) ≥ ν(Y) ≥ μ(Aξ (λ0 )). Since μ is regular and contains no atoms, there exists a Borel set B ⊂ X such that Aξ (λ0 ) ⊆ B ⊆ Ā ξ (λ0 ) and μ(B) = ν(Y). Let μ̄ := μ⌊B be the restriction of μ to B. We leave it to the ̄ reader to verify that μ̄ ≤ μ, μ(X) = ν(Y), and sup

̃ ̃ μ≤μ, μ(X)=ν(Y)

∫ ξdμ̃ = ∫ ξdμ = ∫ ξdμ.̄ X

B

X

Since (ξ , p) ∈ 𝒥 θ , ([ξ − λ0 ]+ , p + λ0 ) ∈ 𝒥 θ as well. Since ∫[ξ − λ0 ]+ dμ = ∫ ξdμ̄ − λ0 ν(Y), ∫(p + λ0 )dν = ∫ pdν + λ0 ν(Y),

X

X

Y

Y

we get (

sup

̃ ̃ μ≤μ, μ(X)=ν(Y)

̃ + ∫ pdν = ∫[ξ − λ0 ]+ dμ + ∫(p + λ0 )dν. ∫ ξdμ)

X

Y

X

Y

Since [ξ − λ0 ]+ ≥ 0 on X and ([ξ − λ0 ]+ , p + λ0 ) ∈ 𝒥 θ , it follows that } { ̃ + ∫ pdν} ≥ inf ∫ ξdμ) ∫ ξdμ + ∫ pdν, {( ̃ sup ̃ (ξ ,p)∈𝒥 θ ,ξ ≥0 (ξ ,p)∈𝒥 θ μ≤μ, μ(X)=ν(Y) X Y X Y } { inf

so θ(μ, ν) ≥

inf

(ξ ,p)∈𝒥 θ ,ξ ≥0

∫ ξdμ + ∫ pdν. X

Y

On the other hand, by (9.6) we get θ(μ, ν) ≤

inf

(ξ ,p)∈𝒥 θ

∫ ξdμ + ∫ pdν ≤ X

Y

inf

(ξ ,p)∈𝒥 θ ,ξ ≥0

∫ ξdμ + ∫ pdν, X

Y

so the equality is verified for θ(μ, ν) in the case μ(X) > ν(Y). The claim follows from Proposition 9.1.

9.3 Deterministic transport The subject of existence (and uniqueness) of a deterministic transport plan plays a major part of the optimal transport literature. Here we only sketch the fundamental ideas, extended as well to unbalanced transport. The existence of optimal deterministic transport is related to the existence of optimizers to the dual problem as given by Theorems 9.1 and 9.2.

122 | 9 Optimal transport for scalar measures Following the current literature in optimal transport (e. g., [48, 41]) we define the transform p ∈ C(Y) → pθ ∈ C(X): pθ (x) = sup θ(x, y) − p(y). y∈Y

(9.10)

Likewise for the transform ξ ∈ C(X) → ξθ ∈ C(Y): ξθ (y) = sup θ(x, y) − ξ (x). x∈X

(9.11)

Note that if X = Y and θ is a symmetric function (θ(x, y) = θ(y, x) ∀(x, y) ∈ X × X), then both definitions are reduced to the same one. In that case, the functions of the form pθ are called θ convex. We shall adopt this notation in the general case. Definition 9.3.1. A function ξ ∈ C(X) is θX convex if ξ = pθ for some p ∈ C(Y). Likewise, p ∈ C(Y) is θY convex if p = ξθ for some ξ ∈ C(X). We denote by ΘX (resp. ΘY ) the set of θX (resp. θY ) convex functions. By the assumed compactness of X, Y and the continuity (hence uniform continuity) of θ, the θ convex functions are always continuous. In particular, we have the following. Proposition 9.2. (i) For any p ∈ C(Y), pθ ∈ C(X) and (pθ , p) ∈ 𝒥 θ . Likewise, for any ξ ∈ C(X), ξθ ∈ C(Y) and (ξ , ξθ ) ∈ 𝒥 θ . (ii) For any p ∈ C(Y) and y ∈ y, pθθ (y) := (pθ )θ (y) ≤ p(y). Likewise, for any ξ ∈ C(X) and x ∈ X, ξθθ (x) := (ξθ )θ (x) ≤ ξ (x). (iii) ξ is θX convex iff ξθθ = ξ . The same implies for θY convex p. (iv) For any θX convex function ξ and any x1 , x2 ∈ X, ξ (x1 ) − ξ (x2 ) ≤ max θ(x1 , y) − θ(x2 , y). y∈Y

Likewise, for any θY convex function p and any y1 , y2 ∈ Y, p(y1 ) − p(y2 ) ≤ max θ(x, y1 ) − θ(x, y2 ). x∈X

Proof. The proof follows directly from the definitions. We shall only present the proof of the only if part in (iii) and leave the rest for the reader. If ξ is θX convex, then there exists p ∈ C(Y) such that ξ = pθ . We show that ξθθ := θθ pθ = pθ ≡ ξ . By definition pθθ sup θ(x, y) − θ(x 󸀠 , y) + θ(x 󸀠 , y󸀠 ) − p(y󸀠 ). θ (x) = sup inf 󸀠 y

x

y󸀠

θ 󸀠 If we substitute y = y󸀠 we get the inequality pθθ θ (x) ≥ p (x). If we substitute x = x we get the opposite inequality.

9.3 Deterministic transport | 123

Proposition 9.2(i) and Theorems 9.1 and 9.2 enable us to reduce the minimization of the dual problem from the set of pairs 𝒥 θ to the set of θ convex functions on either X or Y. Theorem 9.3. If μ(X) = ν(Y), then θ(μ, ν) = inf ∫ ξdμ + ∫ ξθ dν = inf ∫ pθ dμ + ∫ pdν, ξ ∈ΘX

X

p∈ΘY

Y

X

Y

while if μ(X) < ν(Y) and θ ≥ 0, then θ(μ, ν) =

inf

ξ ∈ΘX ;ξ ≥0

∫ ξdμ + ∫[ξθ ]+ dν,

X

Y

and if μ(X) < ν(Y), then θ(μ, ν) =

inf

p∈ΘY ;p≥0

∫[pθ ]+ dμ + ∫ pdν.

X

Y

9.3.1 Solvability of the dual problem Let us start from the balanced case. Let p ∈ C(Y), pθ ∈ C(X). Let y2 ∈ Y be a maximizer in (9.10). Then pθ (x1 ) − pθ (x2 ) ≤ θ(x1 , y2 ) − p(y2 ) − pθ (x2 ) = θ(x1 , y2 ) − p(y2 ) − [θ(x2 , y2 ) − p(y2 )] = θ(x1 , y2 ) − θ(x2 , y2 ) ≤ max |θ(x1 , y) − θ(x2 , y)|. y∈Y

Let us assume that X is a metric compact space and dX is the metric on X. It follows that there exists a continuous, non-negative-valued function σ on ℝ+ such that σ(0) = 0 and max |θ(x1 , y) − θ(x2 , y)| ≤ σ(dX (x1 , x2 )). y∈Y

In particular it follows that for any p ∈ C(Y), pθ is subjected to a modulus of continuity σ determined by θ: |pθ (x1 ) − pθ (x2 )| ≤ σ(dX (x1 , x2 ))

∀x1 , x2 ∈ X.

If we further assume that Y is a compact metric space and dY is the associated metric, we obtain the same result for ξθ , where ξ ∈ C(X) (9.11): |ξθ (y1 ) − ξθ (y2 )| ≤ σ(dX (x1 , x2 ))

∀y1 , y2 ∈ Y.

124 | 9 Optimal transport for scalar measures We may reduce now the sets J θ , J θ in Theorems 9.1 and 9.2 to uniformly bounded and equi-continuous pairs of functions. Moreover, we may assume that the pairs are bounded in supremum norm as well (why?). By the Arzelà–Ascoli theorem we get the uniform convergence of minimizing/maximizing sequence to an optimizer. Thus we replace inf by min and sup by max in Theorems 9.1 and 9.2. In particular, we obtain the following. Lemma 9.1. In the balanced case there exists (ξ0 , p0 ) ∈ 𝒥 θ such that ξ0 = pθ0 , p0 = (ξ0 )θ , and θ(μ, ν) = ∫ ξ0 μ(dx) + ∫ p0 ν(dy). X

(9.12)

Y

If μ(X) > ν(Y), then there exists such a pair for which p0 ∈ C(Y; ℝ+ ), while if μ(X) < ν(Y), then ξ0 ∈ C(X; ℝ+ ). Lemma 9.2. Any optimal plan π0 for θ(μ, ν) is supported in the set {(x, y) : ξ0 (x)+p0 (y) = θ(x, y)}. Proof. By Theorems 9.1 and 9.2 and Lemma 9.1 it follows that if π0 is optimal, then θ(μ, ν) = ∫ θ(x, y)π0 (dxdy) = ∫ ξ0 dμ + ∫ p0 dν. X×Y

X

Y

In the balanced case, we get π0 ∈ Π(μ, ν), so ∫ ∫ ξ0 (x) + p0 (y) − θ(x, y)π0 (dxdy) = 0. X Y

Since ξ0 (x) + p0 (y) ≥ θ(x, y), we get the claim for the balanced case. In the unbalanced case μ(X) < ν(Y), let μ̃ ≤ μ be the X marginal of π0 . Then ∫ ξ0 dμ + ∫ p0 dν = ∫ ξ0 dμ̃ + ∫ p0 dν + ∫ ξ0 (dμ − dμ)̃ X

Y

X

Y

X

= ∫ [ξ0 (x) + p0 (y)]dπ0 + ∫ ξ0 (dμ − dμ)̃ ≥ θ(μ, ν), X×Y

X

where the last inequality follows from ξ0 (x)+p0 (y) ≥ θ(x, y) and ξ0 ≥ 0 via Theorem 9.2. It implies again that the support of π0 is contained in the set {(x, y) : ξ0 (x) + p0 (y) = θ(x, y)} and, in addition, that ξ0 = 0 on the support of μ − μ.̃ An analogous argument applies for the case μ(X) > ν(Y). We now sketch the way to obtain existence and uniqueness of a deterministic transport map π0 . For this we replace the assumption that X, Y are compact sets by X = Y = ℝd , but supp(μ), supp(ν) are compact subsets in ℝd . In addition we assume that θ ∈ C 1 (ℝd × ℝd ) and the function y → ∇x θ(x, y) is injective for any x, i. e., ∇x θ(x, y1 ) = ∇x θ(x, y2 ) ⇒ y1 = y2 .

(9.13)

9.4 Metrics on the set of probability measures | 125

Theorem 9.4. Assume the supports of both μ and ν are bounded in ℝn and that the twist condition (9.13) is satisfied for θ. Let (ξ0 , p0 ) be the dual pair verifying (9.12). Then ξ0 is differentiable μ-a. e. and there exists a measurable mapping T on supp(μ) verifying ∇x θ(x, y) = ∇x ξ0 (x) μ-a.e., where y = T(x). Moreover, any optimal plan π ∈ Π(μ, ν) of u(μ, ν) is supported in the graph of T: sup(π) ⊂ {(x, y); y = T(x)} . In particular, such T satisfies T# μ = ν, that is, μ(T −1 (B)) = ν(B) ∀B ⊂ Y measurable,

(9.14)

and this mapping is the solution of the Monge problem max ∫ θ(x, S(x))dμ.

S# μ=ν

(9.15)

Sketch of proof. Let (ξ0 , p0 ) ∈ 𝒥 θ be the optimal solution of the dual problem. Assuming (x, y) ∈ supp(π0 ), by Lemma 9.2 we get that the equality ξ0 (x) + p0 (y) = θ(x, y), while ξ0 (z) + p0 (y) ≥ θ(z, y) for any z by definition. If ξ0 is differentiable, then this implies ∇x ξ0 (x) = ∇x θ(x, y). By the twist condition (9.13), this determines y and we denote y := T(x).

9.4 Metrics on the set of probability measures Let us invert maximum to minimum in (9.1). We obtain c(μ, ν) := min ∫ ∫ c(x, y)π(dxdy), π∈Π(μ,ν)

X Y

where c ∈ C(X, ×Y) is now considered as a cost of transportation. This can easily be observed to be equivalent to (9.1), upon choosing c = −θ. In the dual formulation we have to invert the inequality in 𝒥 θ and consider 𝒥 c := {(ξ , p) ∈ C(X) × C(Y); ξ (x) + p(y) ≤ c(x, y) ∀(x, y) ∈ X × Y}.

(9.16)

If we restrict ourselves to the balanced case μ(X) = ν(Y), then Theorem 9.1 takes the form c(μ, ν) = sup ∫ ξdμ + ∫ pdν. (ξ ,p)∈𝒥 c

X

Y

126 | 9 Optimal transport for scalar measures Note, however, that if we assume that c is non-negative (as we did for θ in Assumption 9.1.1), then we have to invert the inequalities in the definition of Π(μ, ν) (9.2) in order to avoid a trivial minimizer π = 0 in the imbalanced case μ(X) ≠ ν(Y) (Remark 9.2.1). In the special case of X = Y = ℝd we may consider cq (x, y) = |x − y|q . Of particular interest is the case q ≥ 1, which leads to the definition of metrics on the set of probability measures on ℝd of finite q moment: (q)

ℳ1

:= {μ ∈ ℳ1 (ℝd ), ∫ |x|q dμ < ∞}.

(9.17)

Indeed, it turns out that Wq (μ, ν) := cq (μ, ν)1/q

(9.18)

is a metric on ℳ(q) 1 , called (perhaps unjustly, see [47]) the Wasserstein metric [9, 48]. 9.4.1 Special cases Example 9.4.1. Suppose θ(x, y) = x ⋅ y is the inner product in ℝd . Since |x ⋅ y| ≤ (|x|2 + |y|2 )/2, we get that θ(μ, ν) is bounded on ℳ(2) 1 . The connection with W2 is apparent via (9.18) for q = 2: W22 (μ, ν) :=

inf

π∈Π(μ,ν)

∫ |x − y|2 π(dx) X×X

2

= ∫ |x| μ(dx) + ∫ |x|2 ν(dx) − sup ∫ x ⋅ ydπ π∈Π(μ,ν)

and supπ∈Π(μ,ν) ∫ x⋅ydπ = θ(μ, ν). The definition θ(μ, ν) where θ(x, y) = x⋅y stands for the statistical correlation between random variables distributed according to μ, ν. Thus, the Wasserstein W2 metric is related to the matching of such two random variables with maximal correlation. In this special case θ(x, y) = x ⋅ y corresponding to the Wasserstein metric W2 we get that the optimal mapping T is just the gradient of the function ξo : T(x) = ∇x ξ0 (x).

(9.19)

In a pioneering paper, Brenier [9] considered the quadratic cost function c(x, y) = |x − y|2 , and proved that the optimal potential ξ0 is a convex function. In particular, he showed the following.

9.4 Metrics on the set of probability measures | 127

d Theorem 9.5 ([9]). For any pair of probability measures μ, ν∈ℳ(2) 1 (ℝ ) (equation (9.17)), where μ is absolutely continuous with respect to Lebesgue measure, there exists a unique convex function ξ such that ∇ξ# μ = ν, and

∫ |x − ∇ξ (x)|2 dμ < ∫ |S(x) − x|2 dμ for any S ≠ ∇ξ satisfying S# μ = ν. This result is one of the most quoted papers in the corresponding literature. d Corollary 9.4.1. Let μ ∈ ℳ(2) 1 (ℝ ) be absolutely continuous with respect to Lebesgue measure and let ϕ : ℝd → ℝ be convex. Then T := ∇ϕ is a measurable mapping and ν := ∇ϕ# μ ∈ ℳ(2) 1 . Moreover, T is the only solution of the Monge problem with respect to the cost c(x, y) = |x − y|2 for μ, ν.

Example 9.4.2. Suppose X = Y is a metric space and d is the corresponding metric. The metric Monge distance between μ and ν is defined as d(μ, ν) := min

π∈Π(μ,ν)

∫ d(x, y)π(dxdy). X×X

Let us define θ(x, y) = λ − d(x.y),

(9.20)

where λ ≥ maxx,y∈X d(x, y) (here we take advantage of our assumption that X is a compact space). Thus d(μ, ν) = λ − θ(μ, ν) ≥ 0. Using (9.11), ξθ (y) = max λ − d(x, y) − ξ (x) = λ − min d(x, y) + ξ (x) := λ − ξd (y). x∈X

x∈X

From its definition, ξd (y) = minx∈X d(x, y) + ξ (x) ∈ Lip(1), where Lip(1) is the set of 1-Lipschitz functions ξd (x1 ) − ξd (x2 ) ≤ d(x1 , x2 ),

x1 , x2 ∈ X.

(9.21)

Indeed, if z1 = arg min d(x1 , ⋅) + ξ (⋅), then for any x2 , z ∈ X ξd (x2 ) − ξd (x1 ) ≤ d(x2 , z) − ξ (z) + d(x1 , z1 ) − ξ (z1 ), and by choosing z = z1 we get (9.21). Moreover, we easily observe that Lip(1) is a selfdual space, i. e., ξd = ξ iff ξ ∈ Lip(1). From Theorem 9.3 we have the following.

128 | 9 Optimal transport for scalar measures In the balanced case μ(X ) = ν(X ), d(μ, ν) = sup ∫ ξd(ν − μ), ξ ∈Lip(1)

(9.22)

X

which is the celebrated Kantorovich–Rubinstein dual formulation of the metric Monge problem [48]. In particular, we obtain that d(μ, ν) depends only on μ − ν, and, in this sense, is a norm on the set of probability measures which lift the metric d from the case space X to the set of probability measures on X. Indeed, we may identify d(x, y) with d(δx , δy ). In the unbalanced case μ(X) > ν(X) we use (9.20) and Theorem 9.3 to obtain d(μ, ν) := λν(X) −

inf

ξ ∈Lip(1),ξ ≥0

∫ ξ (x)μ(dx) + [λ − ξ (x)]+ ν(dx),

X

which holds for any λ > maxx,y∈X d(x, y). In particular, we can take λ > maxX ξ so [λ − ξ ]+ = λ − ξ , and obtain d(μ, ν) := −

inf

ξ ∈Lip(1),ξ ≥0

If μ(X ) > ν(X ),

∫ ξ (x)(μ(dx) − ν(dy)) = X

d(μ, ν) =

sup

ξ ∈Lip(1),ξ ≤0

sup

ξ ∈Lip(1),ξ ≤0

∫ ξ (x)(μ(dx) − ν(dy)). X

∫ ξ (x)(μ(dx) − ν(dy)). X

Likewise, μ(X) < ν(X). If μ(X ) < ν(X ),

d(μ, ν) =

sup

ξ ∈Lip(1),ξ ≥0

∫ ξ (x)(μ(dx) − ν(dy)). X

Remark 9.4.1. d is not extended to a norm (and neither a metric) on the set of positive measures. Only its restriction to the probability measures ℳ1 is a norm. Remark 9.4.2. The norm d on ℳ1 is a metrization of the weak* topology introduced in Section 4.6. See Appendix B.3. 9.4.2 McCann interpolation Let T be a measurable mapping in Euclidean space X. Let μ, ν ∈ ℳ1 (X) and ν = T# μ. Define the interpolation of T with the identity I as Ts := (1 − s)I + sT, where s ∈ [0, 1]. This induces an interpolation between μ and ν via T as follows: μ(s) := Ts# μ,

s ∈ [0, 1].

9.4 Metrics on the set of probability measures | 129

Evidently, μ(0) = μ and μ(1) = ν, while μ(s) ∈ ℳ1 (X) for any s ∈ [0, 1]. Suppose now T is the optimal Monge map for μ, ν ∈ ℳ1 f (2) (ℝd ) with respect to the quadratic cost c(x, y) = |x − y|2 . By Theorem 9.5 T = ∇ξ for some convex function ξ . Then Ts = ∇ξs , where ξs (x) = [(1 − s)|x|2 /2 + sξ (x)] is a convex function for any s ∈ [0, 1]. In particular, by Corollary 9.4.1, Ts = ∇ξs is the optimal mapping of μ to μ(s) , that is, W2 (μ, μ(s) ) = √∫ |∇ξs (x) − x|2 dμ. Since ∇ξs (x) − x = s(∇ξ (x) − x), we get W2 (μ, μ(s) ) = s√∫ |∇ξ (x) − x|2 dμ = sW2 (μ, ν).

(9.23)

W2 (ν, μ(s) ) = (1 − s)W2 (μ, ν),

(9.24)

Likewise,

and μ(s) is the only measure which minimizes (1 − s)W22 (μ, λ) + sW22 (ν, λ) over λ ∈ ℳ(2) 1 . This remarkable identity implies that the orbit μ(s) defined in this way is, in fact, a geodesic path in the set ℳ(2) 1 . See [34, 26].

10 Interpolated costs 10.1 Introduction Assume there exist a compact set Z and a pair of functions θ(1) ∈ C(X × Z; ℝ+ ), θ(2) ∈ C(Y × Z; ℝ+ ), such that θ(x, y) := max θ(1) (x, z) + θ(2) (y, z).

(10.1)

z∈Z

It is more natural, in the current context, to invert the point of view from utility (which should be maximized) to a cost (which should be minimized). Indeed, this is what we did in Section 9.4, and there is nothing new about it whatsoever. All we need is to define the cost c(x, y) = −θ(x, y) and replace maximum by minimum and vice versa. In particular, (10.1) is replaced by c(x, y) := min c(1) (x, z) + c(2) (y, z). z∈Z

(10.2)

Example 10.1.1. If X = Y = Z is a compact convex set in ℝd , r ≥ 1, then c(1) (x, i) = 2r−1 |x − z|r , c(2) (y, i) = 2r−1 |y − z|r verifies (10.2) for c(x, y) = |x − y|r . If r > 1, then the maximum is obtained at the mid-point z = (x + y)/2, and if r = 1 it is obtained at any point in the interval τx + (1 − τ)y, τ ∈ [0, 1]. More generally, if α > 0, then cα(1) (x, i) =

(1 + α1/(r−1) )r |x − z|r , α + αr/(r−1)

θα(2) (y, i) =

α(1 + α1/(r−1) )r |y − z|r , α + αr/(r−1)

which reduces the previous case if α = 1. Example 10.1.2. Let X be a compact Riemannian manifold and let l = l(x, v) be a Lagrangian function on the tangent space (x, v) ∈ 𝕋X, that is, – l ∈ C(𝕋X), – l is strictly convex on the fiber v for (x, v), – l is superlinear in each fiber, i. e., lim‖v‖→∞ l(x,v) = ∞ for any x ∈ X. ‖v‖ For any T > 0 define θT : X × X → ℝ as the minimal action 1

c(x, y) := cT (x, y) :=

min

w∈C 1 (0,T;X),w(0)=x,w(1)=y

̇ ∫ l(w(t)w(t))dt. 0

Then, for any 0 < T1 < T, c(x, y) = min cT1 (x, z) + cT−T1 (y, z), z∈X

so by definition with c = cT we get c(1) (x, z) = cT1 (x, z) and c(2) (y, z) = cT−T1 (y, z). https://doi.org/10.1515/9783110635485-010

132 | 10 Interpolated costs Note that Example 10.1.1 is, indeed, a special case of Example 10.1.2, where l(x, v) := ‖v‖r and T = 1. More generally, we can extend Example 10.1.1 to a geodesic space X where d : X × X → ℝ is the corresponding metric: dr (x, y) = min z∈X

(1 + α1/(r−1) )r r α(1 + α1/(r−1) )r r d (x, z) + d (y, z). r/(r−1) α+α α + αr/(r−1)

(10.3)

10.1.1 Semi-finite approximation: the middle way Let Zm := {z1 , . . . , zm } ⊂ Z be a finite set. Denote by cZm (x, y) := min c(1) (x, zi ) + c(2) (zi , y) ≥ c(x, y) 1≤i≤m

(10.4)

the (Zm ) semi-finite approximation of c given by (10.2). Z The Kantorovich lifting of cm to the set of measures is given by cZm (μ, ν) :=

inf

π∈Π(μ,ν)

∫ cZm (x, y)π(dxdy).

(10.5)

X×Y

An advantage of the semi-discrete method described above is that it has a dual formulation which converts the optimization (10.5) to a convex optimization on ℝm . Indeed, we prove that for a given Zm ⊂ Z there exists a concave function Ξνμ,Zm : ℝm → ℝ such that maxm Ξνμ,Zm (p)⃗ = cZm (μ, ν), ⃗ p∈ℝ

(10.6)

and, under some conditions on either μ or ν, the maximizer is unique up to a uniform translation p⃗ → p⃗ + β(1, . . . , 1) on ℝm . Moreover, the maximizers of Ξνμ,Zm yield a unique congruent optimal partition. The accuracy of the approximation of c(x, y) by cZm (x, y) depends, of course, on the choice of the set Zm . In the special (but interesting) case X = Y = Z = ℝd and c(x, y) = |x − y|q , q > 1 it can be shown that, given a compact set K ⊂ ℝd , for a fairly good choice of Zm ⊂ K one may get cZm (x, y) − c(x, y) = O(m−q/d ) for any x, y ∈ K. From (10.4) and the above reasoning we obtain in particular cZm (μ, ν) − c(μ, ν) ≥ 0

(10.7)

for any pair of probability measures and that, for a reasonable choice of Zm , (10.7) is of order m−2/d if the supports of μ, ν are contained in a compact set. For a given m ∈ ℕ and a pair of probability measures μ, ν, the optimal choice of Zm is the one which minimizes (10.7). Let ϕm (μ, ν) := inf cZm (μ, ν) − c(μ, ν) ≥ 0, Zm ⊂Z

(10.8)

10.2 Optimal congruent partitions | 133

where the infimum is over all sets of m points in Z. Note that the optimal choice now depends on the measures μ, ν themselves (and not only on their supports). A natural question is then to evaluate the asymptotic limits ̄ ν) := lim sup m2/d ϕm (μ, ν); ϕ(μ, m→∞

ϕ(μ, ν) := lim inf m2/d ϕm (μ, ν). m→∞

Some preliminary results regarding these limits are discussed in this chapter.

10.2 Optimal congruent partitions Definition 10.2.1. Given a pair of probability measures μ ∈ ℳ1 (X), ν ∈ ℳ1 (Y) and m ∈ ℕ, a weak congruent m-partition of (X, Y) subject to (μ, ν) is a pair of weak partitions μ⃗ := (μ1 , . . . , μm ), ν⃗ := (ν1 , . . . , νm ), where μi ∈ ℳ+ (X), νi ∈ ℳ+ (Y) such that μi (X) = νi (Y),

1 ≤ i ≤ m.

The set of all weak congruent m-partitions is denoted by 𝒮𝒫 w μ,ν (m). Since, by assumption, neither μ nor μ contains atoms it follows that 𝒮𝒫 w μ,ν (m) ≠ 0 for any m ∈ ℕ. Lemma 10.1. We have cZm (μ, ν) =

min

∑ [∫ c(1) (x, zi )μi (dx) + ∫ c(2) (y, zi )νi (dy)] , Y [X ]

w ⃗ (μ,⃗ ν)∈𝒮𝒫 μ,ν (m) 1≤i≤m

where cZm (μ, ν) is as defined by (10.5) and (μ,⃗ ν)⃗ ∈ 𝒮𝒫 w μ,ν (m). Proof. First note that the existence of minimizer follows by compactness of the measures in the weak* topology (Section 4.6.1). Define, for 1 ≤ i ≤ m, Γi := {(x, y) ∈ X × Y; c(1) (x, zi ) + c(2) (y, zi ) = cZm (x, y)} ⊂ X × Y. Note that, in general, the choice of {Γi } is not unique. However, we may choose {Γi } as measurable, pairwise disjoint sets in X × Y. Given π ∈ ΠYX (μ, ν), let πi be the restriction of π to Γi . In particular, ∑1≤i≤m πi = π. Let μi be the X marginal of πi and νi the y marginal of πi . Then (μ,⃗ ν)⃗ defined in this way Zm (1) (2) is in 𝒮𝒫 w μ,ν (m). Since by definition c (x, y) = c (x, zi ) + c (y, zi ) πi -a. s., ∫ ∫ cZm (x, y)π(dxdy) = ∑ ∫ ∫ cZm (x, y)πi (dxdy) X Y

1≤i≤m X Y

= ∑ ∫ ∫ c(1) (x, zi )πi (dxdy) + ∫ c(2) (y, zi )πi (dxdy) 1≤i≤m X Y

X

134 | 10 Interpolated costs

= ∑ [∫ c(1) (x, zi )μi (dx) + ∫ c(2) (y, zi )νi (dy)] . 1≤i≤m [X Y ]

(10.9)

Choosing π above to be the optimal transport plan we get the inequality cZm (μ, ν) ≥

∑ [∫ c(1) (x, zi )μi (dx) + ∫ c(2) (y, zi )νi (dy)] . Y [X ]

inf

w ⃗ (μ,⃗ ν)∈𝒮𝒫 μ,ν (m) 1≤i≤m

To obtain the opposite inequality, let (μ,⃗ ν)⃗ ∈ 𝒮𝒫 w μ,ν (m) and set ri := μi (X) ≡ νi (Y). Define π(dxdy) = ∑1≤i≤m ri−1 μi (dx)νi (dy). Then π ∈ ΠYX (μ, ν) and from (10.4) ∫ ∫ cZm (x, y)π(dxdy) = ∑ ∫ ∫ cZm (x, y)ri−1 μi (dx)νi (dy) 1≤i≤m X Y

X Y

≤ ∑ ∫(c(1) (x, zi ) + c(2) (y, zi ))ri−1 μi (dx)νi (dy) 1≤i≤m X

= ∑ [∫ c(1) (x, zi )μi (dx) + ∫ c(2) (y, zi )νi (dy)] 1≤i≤m [X Y ]

(10.10)

and we get the second inequality. Given p⃗ = (pz1 , . . . , pzm ) ∈ ℝm , let ξZ(1) (p,⃗ x) := min c(1) (x, zi ) + pi ; ξZ(2) (p,⃗ y) := min c(2) (y, zi ) + pi , m

1≤i≤m

1≤i≤m

m

(10.11)

ΞZμm (p)⃗ := ∫ ξZ(1) (p,⃗ x)μ(dx); ΞZν m (p)⃗ := ∫ ξZ(2) (p,⃗ y)ν(dy),

(10.12)

m ⃗ ΞZμ,ν (p)⃗ := ΞZμm (p)⃗ + ΞZν m (−p).

(10.13)

X

m

Y

m

For any r ⃗ in the simplex Δ̄ m (1) let (−ΞZμm )∗ (−r)⃗ := sup ΞZμm (p)⃗ − p⃗ ⋅ r.⃗

(10.14)

⃗ m p∈ℝ

Analogously, for ν ∈ ℳ1 (Y) (−ΞZν m )∗ (−r)⃗ := sup ΞZν m (p)⃗ − p⃗ ⋅ r.⃗

(10.15)

⃗ m p∈ℝ

Compare these with the function Ξ+ in Section 4.4. Lemma 10.2. We have (−ΞZμm )∗ (−r)⃗ = c(1) (μ, ∑ ri δzi ) , 1≤i≤m

(−ΞZν m )∗ (−r)⃗ = c(2) (ν, ∑ δz ) . 1≤i≤m

10.2 Optimal congruent partitions | 135

Proof. This is a special case (for the scalar case J = 1) of the partition problems discussed in Section 7.1. See also [48]. It is also a special case of generalized partitions; see Theorem 3.1 and its proof in [50]. Theorem 10.1. We have m sup ΞZμ,ν (p)⃗ = cZm (μ, ν).

(10.16)

⃗ m p∈ℝ

Proof. From Lemma 10.1, Lemma 10.2, and Definition 10.2.1 we obtain ⃗ . cZm (μ, ν) = inf [(−ΞZμm )∗ (−r)⃗ + (−ΞZν m )∗ (−r)] ⃗ Δ̄ m (1) r∈

Z

(10.17)

Z

Note that (−Ξμm )∗ , (−Ξν m )∗ as defined in (10.14), (10.15) are, in fact, the Legendre transZ

Z

forms of −Ξμm , −Ξν m , respectively. As such, they are defined formally on the whole domain ℝm (considered as the dual of itself under the canonical inner product). It follows Z Z that (−Ξμm )∗ (r)⃗ = (−Ξν m )∗ (r)⃗ = ∞ for r ⃗ ∈ ℝm − Δm (1). Note that this definition is consism (2) tent with the right-hand side of (10.14), (10.15), since c(1) (μ, ∑m 1 ri δzi ) = c (ν, ∑1 ri δzi ) = m m ∞ if ∑i=1 ri δzi is not a probability measure, i. e., r ⃗ ∈ ̸ Δ (1). Z

Z

On the other hand, Ξμm and Ξν m are both finite and continuous on the whole of ℝ . The Fenchel–Rockafellar duality theorem ([48], Theorem 1.9) then implies m

⃗ sup ΞZμm (p)⃗ + ΞZν m (−p)⃗ = infm (−ΞZμm )∗ (r)⃗ + (−ΞZν m )∗ (r). ⃗ r∈ℝ

⃗ m p∈ℝ

(10.18)

The proof follows from (10.13), (10.17). An alternative proof We can prove (10.16) directly by constrained minimization, as follows: (μ,⃗ ν)⃗ ∈ 𝒮𝒫 w μ,ν (m) iff F(p,⃗ ϕ, ψ) := ∑ pi (∫ dμi − ∫ dνi ) + ∫ ϕ(x) (μ(dx) − ∑ μi (dx))

1≤i≤m

X

Y

X

1≤i≤m

+ ∫ ψ(y) (ν(dy) − ∑ νi (dy)) ≤ 0 1≤i≤m

Y

for any choice of p⃗ ∈ ℝm , ϕ ∈ C(X), ψ ∈ C(Y). Moreover, supp,ϕ,ψ F = ∞ unless ⃗ w Zm (μ,⃗ ν)⃗ ∈ 𝒮𝒫 μ,ν (m). We then obtain from Lemma 10.1 that c (μ, ν) = inf

sup

∑ [∫ c(1) (x, zi )μi (dx) + ∫ c(2) (y, zi )νi (dy)] 1≤i≤m [X Y ] + F(p,⃗ ϕ, ψ)

{μi ∈ℳ+ (X),νi ∈ℳ+ (Y)} p∈ℝ ⃗ m ,ϕ∈C(X),ψ∈C(Y)

136 | 10 Interpolated costs

=

sup

inf

∑ ∫ (c(1) (x, zi ) + pi − ϕ(x)) μi (dx)

⃗ m ,ϕ∈C(X),ψ∈C(Y) {μi ∈ℳ+ (X),νi ∈ℳ+ (Y)} 1≤i≤m p∈ℝ

X

+ ∑ ∫ (c(2) (y, zi ) − pi − ψ(y)) νi (dy) + ∫ ϕμ(dx) + ∫ ψν(dy). 1≤i≤m Y

X

(10.19)

Y

We now observe that the infimum on {μi , νi } above is −∞ unless c(1) (x, zi )+pi −ϕ(x) ≥ 0 and c(2) (y, zi ) + pi − ψ(y) ≥ 0 for any 1 ≤ i ≤ m. Hence, the two sums on the right of (10.19) are non-negative, so the infimum with respect to {μi , νi } is zero. To obtain the supremum on the last two integrals on the right of (10.19) we choose ϕ, ψ as large as possible under this constraint, namely, ϕ(x) = min c(1) (x, zi ) + pi , 1≤i≤m

ψ(y) = min c(2) (y, zi ) − pi , 1≤i≤m

so ϕ(x) ≡ ξZ(1) (p,⃗ x), ψ(y) ≡ ξZ(2) (−p,⃗ y) by definition via (10.11). m

m

10.3 Strong partitions We now define strong partitions as a special case of weak congruent m-partitions (Definition 10.2.1). Definition 10.3.1. Given a pair of probability measures μ ∈ ℳ1 (X), ν ∈ ℳ1 (Y) and m ∈ ℕ, a weak congruent m-partition of (X, Y) subject to (μ, ν) is a pair of strong partitions A⃗ := (A1 , . . . , Am ), B⃗ := (B1 , . . . , Bm ), where Ai ⊂ X, Bi ⊂ Y are measurable strong partitions of X, Y, correspondingly, such that μ(Ai ) = νi (Bi ) 1 ≤ i ≤ m. The set of all strong congruent m-partitions is denoted by 𝒮𝒫 μ,ν (m). Assumption 10.3.1. We have: a) μ(x; c(1) (x, zi ) − c(1) (x, z 󸀠 ) = p) = 0 for any p ∈ ℝ and any zi , zi󸀠 ∈ Zm , b) ν(y; c(2) (y, zi ) − c(2) (z 󸀠 , y) = p) = 0 for any p ∈ ℝ and any zi , zi󸀠 ∈ Zm . Let us also define, for p⃗ ∈ ℝm , Ai (p)⃗ := {x ∈ X; c(1) (x, zi ) + pi = ξZ(1) (p,⃗ x)}, m

Bi (p)⃗ := {y ∈ Y; c(2) (y, zi ) + pi = ξZ(2) (p,⃗ y)}. m

(10.20)

Note that, by (10.11), (10.12) ΞZμm (p)⃗ = ∑

1≤i≤m

∫ (c(1) (x, zi ) + pi )μ(dx); Ai (p)⃗

(10.21)

10.3 Strong partitions | 137

likewise, ΞZν m (p)⃗ = ∑

1≤i≤m

∫ (c(2) (y, zi ) + pi )ν(dy).

(10.22)

Bi (p)⃗

The following lemma is a special case of Lemma 4.3 in [50]. Lemma 10.3. Under Assumption 10.3.1 (a) (resp. (b)), ⃗ (resp. {Bi (p)}) ⃗ induces essentially disjoint partitions of X i) for any p⃗ ∈ ℝm , {Ai (p)} (resp. Y); Z Z ii) Ξμm (resp. Ξν m ) are continually differentiable functions on ℝm , Z

𝜕Ξμm 𝜕pi

⃗ = μ(Ai (p))

resp.

Z

𝜕Ξν m ⃗ = ν(Bi (p)). 𝜕pi

Theorem 10.2. Under Assumption 10.3.1 there exists a unique minimizer r0⃗ of (10.17). Zm In addition, there exists a maximizer p⃗ 0 ∈ ℝm of Ξμ,ν , and {Ai (p⃗ 0 ), Bi (−p⃗ 0 )} induces a unique, strong congruent corresponding partition in X, Y satisfying μ(Ai ) = ν(Bi ) := r0,i , and m

π0 (dxdy) := ∑(r0,i )−1 1Ai (p⃗ 0 ) (x)1Bi (−p⃗ 0 ) (y)μ(dx)ν(dy) 1

(10.23)

is the unique optimal transport plan for cZm (μ, ν). Z

m Proof. The proof is based on the differentiability of Ξμ,ν via Lemma 10.3 and Proposition A.10. See the proof of Theorem 11.1 for details. To prove that π0 given by (10.23) is an optimal plan, observe that π0 ∈ Π(μ, ν), and hence

cZm (μ, ν) ≤ ∫ ∫ cZm (x, y)π0 (dxdy). X Y

Then we get from (10.4) cZm (μ, ν) ≤ ∫ ∫ cZm (x, y)π0 (dxdy) ≤ ∑

1≤i≤m

X Y

= ∑ ( ∫ c(1) (x, zi )μ(dx) + 1≤i≤m

Ai (p⃗ 0 )

∫

(c(1) (x, zi )μ(dx) + c(2) (y, zi )ν(dy))

Ai (p⃗ 0 )×Bi (−p⃗ 0 )

∫

m c(2) (y, zi )ν(dy)) = ΞZμ,ν (p⃗ 0 ) ≤ cZm (μ, ν),

Bi (−p⃗ 0 )

where the last equality is from Theorem 10.1. In particular, the first inequality is an equality so π0 is an optimal plan indeed.

138 | 10 Interpolated costs

10.4 Pricing in hedonic market In adaptation to the model of hedonic market [13] there are three components: The space of consumers (say, X), the space of producers (say, Y), and the space of commodities, which we take here to be a finite set Zm := {z1 , . . . , zm }. The function c(1) := c(1) (x, zi ) is the negative of the utility of commodity 1 ≤ i ≤ m to consumer x, while c(2) := c(2) (y, zi ) is the cost of producing commodity 1 ≤ i ≤ m by producer y. Let μ be a probability measure on X representing the distribution of consumers, and let ν be a probability measure on Y representing the distribution of the producers. Following [13] we add the “null commodity” z0 and assign the zero utility and cost c(1) (x, z0 ) = c(2) (z0 , y) ≡ 0 on X (resp. Y). We understand the meaning that a consumer (producer) chooses the null commodity is that he/she avoids consuming (producing) any item from Zm . The objective of hedonic pricing is to find equilibrium prices for the commodities which will balance supply and demand: Given a price pi for z, consumer x will buy commodity z which minimizes its loss c(1) (x, zi ) + pi , or will buy nothing (i. e., “buy” the null commodity z0 ) if min1≤i≤m c(1) (x, zi )+pi > 0, while producer y will prefer to produce commodity z which maximizes its profit −c(2) (y, zi ) + pi , or will produce nothing if max1≤i≤m −c(2) (y, zi ) + pi < 0. Using notation (10.11)–(10.13) we define ξX0 (p,⃗ x) := min{ξZ(1) (p,⃗ x), 0}; ξY0 (p,⃗ y) := min{ξZ(2) (p,⃗ y), 0}, m

Ξ0μ (p)⃗

:=

m

∫ ξX0 (p,⃗ x)μ(dx);

X

⃗ Ξ0,ν μ (p)

:=

Ξ0ν (p)⃗

Ξ0μ (p)⃗

+

:=

∫ ξY0 (p,⃗ y)ν(dy),

Y 0 ⃗ Ξν (−p).

⃗ is the difference between the total loss of all consumers and the total Thus, Ξ0,ν μ (p) profit of all producers, given the price vector p.⃗ It follows that an equilibrium price vector balancing supply and demand is the one which (somewhat counter-intuitively) maximizes this difference. The corresponding optimal strong m-partition represents the matching between producers of (Bi ⊂ Y) to consumers (Ai ⊂ X) of z ∈ Z. The introduction of null commodity allows the possibility that only part of the consumer (producer) communities actually consume (produce), that is, ⋃1≤i≤m Ai ⊂ X and ⋃1≤i≤m Bi ⊂ Y, with A0 = X − ⋃1≤i≤m Ai (B0 = Y − ⋃1≤i≤m Bi ) being the set of non-buyers (nonproducers). Z From the dual point of view, an adaptation c0m (x, y) := min{cZm (x, y), 0} of (10.4) (in the presence of null commodity) is the cost of direct matching between producer y and consumer x. The optimal matching (Ai , Bi ) is the one which minimizes the total Z cost c0m (μ, ν) over all congruent subpartitions as defined in Definition 10.3.1, with the possible inequality μ(∪Ai ) = ν(∪Bi ) ≤ 1.

10.5 Dependence on the sampling set | 139

10.5 Dependence on the sampling set So far we considered the sampling set Zm ⊂ Z as a fixed set. Now we consider the effect of optimizing Zm within the sets of cardinality m in Z. As we already know (equation (10.4)), cZm (x, y) ≥ c(x, y) on X × Y for any (x, y) ∈ X × Y and Zm ⊂ Z. Hence also cZm (μ, ν) ≥ c(μ, ν) for any μ, ν ∈ ℳ1 and any Zm ⊂ Z as new well. An improvement of Zm is a new choice Zm ⊂ Z of the same cardinality m such new Zm Zm that c (μ, ν) < c (μ, ν). In Section 10.5.1 we propose a way to improve a given Zm ⊂ Z, once the optimal partition is calculated. Of course, the improvement depends on the measure μ, ν. In Section 10.5.2 we discuss the limit m → ∞ and prove some asymptotic estimates. 10.5.1 Monotone improvement Proposition 10.1. Define Ξνμ,Zm on ℝm with respect to Zm := {z1 , . . . , zm } ∈ Z as in (10.13). Zm Let (μ,⃗ ν)⃗ ∈ 𝒮𝒫 w μ,ν (m) be the optimal partition corresponding to c (μ, ν). Let ζ (i) ∈ Z be a minimizer of Z ∋ ζ 󳨃→ ∫ c(1) (x, ζ )μzi (dx) + ∫ c(2) (ζ , y)νzi (dy). X

(10.24)

Y

new Let Zm := {ζ (1), . . . , ζ (m)}. Then c

new Zm

(μ, ν) ≤ cZm (μ, ν). ν,Z

Corollary 10.5.1. Let Assumption 10.3.1(a), (b) hold, and let p⃗ 0 be the minimizer of Ξμ m in ℝm . Let {Ai (p⃗ 0 ), Bi (−p⃗ 0 )} be the strong partition corresponding to Zm as in (10.20). new Then the components of Zm are obtained as the minimizers of Z ∋ ζ 󳨃→ ∫ c(1) (x, ζ )μ(dx) + Ai (p⃗ 0 )

c(2) (ζ , y)ν(dy).

∫ Bi (−p⃗ 0 )

new Proof of Proposition 10.1. Let Ξν,new be defined with respect to Zm . By Lemma 10.1 μ ν,Zm

and Theorem 10.1, Ξν,new (p)⃗ ≤ Ξνμ (p⃗ ∗ ) := maxp∈ℝ ⃗ m Ξμ μ

maxℝm Ξν,new (p)⃗ ≡ c μ

new Zm

ν,Zm

(μ, ν) ≤ maxp∈ℝ ⃗ m Ξμ

(p)⃗ ≡ cZm (μ, ν).

(p)⃗ for any p⃗ ∈ ℝm , so

Remark 10.5.1. If X = Y = Z is a Euclidean space and c(x, y) = |x − y|2 , then z new is the center of mass of (Ai (p⃗ 0 ), μ) and (Bi (−p⃗ 0 ), ν): z new :=

∫A (p⃗ ) xμ(dx) + ∫B (−p⃗ ) yν(dy) i

0

i

0

μ(Ai (p⃗ 0 )) + ν(Bi (−p⃗ 0 ))

Let cm (μ, ν) :=

inf

Zm ⊂Z; #(Zm )=m

cZm (μ, ν).

.

140 | 10 Interpolated costs k k k Let Zm := {z1k , . . . , zm } ⊂ Z be a sequence of sets such that zik+1 is obtained from Zm via (10.24). Then by Proposition 10.1 k+1

k

0

c(μ, ν) ≤ cm (μ, ν) ≤ ⋅ ⋅ ⋅ ≤ cZm (μ, ν) ≤ cZm (μ, ν) ≤ ⋅ ⋅ ⋅ ≤ cZm (μ, ν). Open problem. Under which additional conditions may one guarantee k

lim cZm (μ, ν) = cm (μ, ν)?

k→∞

10.5.2 Asymptotic estimates Recall the definition (10.8), ϕm (μ, ν) := inf cZm (μ, ν) − c(μ, ν) := cm (μ, ν) − c(μ, ν) ≥ 0. Zm ⊂Z

Consider the case X = Y = Z = ℝd and c(x, y) = min h(|x − z|) + h(|y − z|), z∈ℝd

where h : ℝ+ → ℝ+ is convex, monotone increasing, twice continuous differentiable. Lemma 10.4. Suppose both μ and ν are supported in a compact set K ⊂ ℝd . Then there exists D(K) < ∞ such that lim sup m2/d ϕm (μ, ν) ≤ D(K). m→∞

(10.25)

Proof. By Taylor expansion of z → h(|x − z|) + h(|y − z|) at z0 = (x + y)/2 we get h(|x − z|) + h(|y − z|) = 2h(|x − y|/2) +

|x − y| 1 ) [(x − y) ⋅ (z − z0 )]2 + o2 (z − z0 ). h󸀠󸀠 ( 2 2|x − y|2

Let now Zm be a regular grid of m points which contains the support K. The distance between any z ∈ K to the nearest point in the grid does not exceed C(K)m−1/d , for some constant C(K). Hence cm (x, y) − c(x, y) ≤ sup |h󸀠󸀠 |C(K)2 m−2/d if x, y ∈ K. Let π0 (dxdy) be the optimal plan corresponding to μ, ν, and c. Then, by definition, c(μ, ν) = ∫ ∫ c(x, y)π0 (dxdy); X Y

cm (μ, ν) ≤ ∫ ∫ cm (x, y)π0 (dxdy), X Y

so ϕm (μ, ν) ≤ ∫ ∫(cm (x, y) − c(x, y))π0 (dxdy) ≤ sup |h󸀠󸀠 |C(K)2 m−2/d , X Y

since π0 is a probability measure.

10.5 Dependence on the sampling set | 141

If h(s) = 2q−1 sq (hence c(x, y) = |x − y|q ), then the condition of Lemma 10.4 holds if q ≥ 2. Note that if μ = ν, then c(μ, μ) = 0, so ϕm (μ, μ) = infZm ∈Z cZm (μ, μ). In that particular case we can improve the result of Lemma 10.4 using Zador’s theorem for vector quantization. Theorem 10.3 ([21, 54]). Let f ∈ 𝕃1 (ℝd ) be a density (with respect to Lebesgue) of a probability measure (in particular f ≥ 0 and ∫ℝd f = 1). Then lim m

n→∞

q/d

(d+q)/d

q

min ∫ min |x − z| f (z)dz = Cd,q [ ∫ f z∈Zm [ℝd ℝd

Zm ⊂ℝd

d/(d+q) ]

.

]

Corollary 10.5.2. If c(x, y) = |x − y|q , q ≥ 2, X = Y = Z = ℝd , and ν = μ = f (x)dx, (d+q)/d

lim mq/d ϕm (μ, μ) = 2q Cd,q (∫ f d/(d+q) dx)

m→∞

,

(10.26)

where Cd,q is some universal constant. Zm Z Zm ⃗ ⃗ is an even function. Hence its maximizer Proof. From (10.13), Ξμ,μ (p)⃗ = Ξμm (p)+Ξ μ (−p) ⃗ must be p = 0. By Theorem 10.1, m ΞZμ,μ (0) = 2ΞZμm (0) = cZm (μ, μ).

(10.27)

Using (10.11), (10.12) with c(1) (x, y) = c(2) (y, x) = 2q−1 |x − y|q , we get ΞZμm (0) = 2q−1 ∫ min |x − z|q μ(dx). ℝd

1≤i≤m

(10.28)

Let now μ = fdx. Since, evidently, c(μ, μ) = 0, from (10.27), (10.28), and Theorem 10.3 we get (10.26). Note that Corollary 10.5.2 does not contradict Lemma 10.4. In fact q ≥ 2 is compatible with the lemma, and (10.25) holds with D(K) = 0 if q > 2. If q ∈ [1, 2), however, then the condition of the lemma is not satisfied (as h󸀠󸀠 is not bounded near 0), and the proposition is a genuine extension of the lemma, in the particular case μ = ν. In the particular case q = 2 we can extend Corollary 10.5.2 to the general case μ ≠ ν, under certain conditions. Let X = Y = Z = ℝd , c(x, y) = |x − y|2 , μ, ν ∈ ℳ(2) 1 (recall (9.18)). Assume μ, ν are absolutely continuous with respect to Lebesgue measure on ℝd . In that case, the Brenier polar factorization theorem, Theorem 9.5, implies the existence of a unique solution to the quadratic Monge problem, i. e., a Borel mapping T such that T# μ = ν (9.19). Let λ be the McCann interpolation between μ and ν corresponding to the middle point s = 1/2 (Section 9.4.2). It turns out that λ is absolutely continuous with respect to Lebesgue measure ℒ as well. Let f := dλ/dℒ ∈ 𝕃1 (ℝd ).

142 | 10 Interpolated costs Theorem 10.4. Under the above assumptions, (d+2)/d

lim sup m2/d ϕm (μ, ν) ≤ 4Cd,2 (∫ f d/(d+2) dx) m→∞

.

Proof. Let S1 be the Monge mapping transporting λ to μ, and let S2 be the Monge mapping transporting λ to ν. In particular μ = S1# λ, ν = S2,# λ and (recall c(⋅, ⋅) := W22 (⋅, ⋅)) we get by (9.24), (9.23) c(λ, μ) = ∫ |S1 (z) − z|2 dλ = c(λ, ν) = ∫ |S2 (z) − z|2 dλ =

1 c(μ, ν). 4

(10.29)

Given 1 ≤ i ≤ m, let λ1 , . . . , λm be a weak m-partition of Λ. In particular ∑m 1 λi = λ. Let zi be the center of mass of λi , so ∫ zdλi = λi (ℝd )zi .

(10.30)

c(μ, ν) = 2 [ ∑ ∫ |S1 (z) − z|2 dλi + ∑ ∫ |S2 (z) − z|2 dλi ] .

(10.31)

From (10.29) it follows that

1≤i≤m

1≤i≤m

Let μi := S1,# λi , νi := S2,# λi . In particular, νi (ℝd ) = μi (ℝd ) = λi (ℝd ), so {μi }, {νi } is a congruent weak partition (Definition 10.2.1). From Lemma 10.1, cZm (μ, ν) ≤ 2 ( ∑ ∫ |x − zi |2 μi (dx) + ∑ ∫ |y − zi |2 νi (dy)) 1≤i≤m

1≤i≤m

2

= 2 ∑ ∫ {|S1 (z) − zi | + |S2 (z) − zi |2 } dλi . 1≤i≤m

Hence (refer to (10.8)) ϕZm (μ, ν) ≤ 2 ∑ ∫ [|S1 (z) − zi |2 − |S1 (z) − z|2 ] dλ 1≤i≤m V

i

+ 2 ∑ ∫ [|S2 (z) − zi |2 − |S2 (z) − z|2 ] dλ. 1≤i≤m V

i

Using the identity |Sκ (z) − zi |2 − |Sκ (z) − z|2 = |zi |2 − |z|2 − 2Sκ (z) ⋅ (zi − z) for κ = 1, 2 we get |S1 (z) − zi |2 − |S1 (z) − z|2 + |S2 (z) − zi |2 − |S2 (z) − z|2

= 2|zi |2 − 2|z|2 − 2(S1 (z) + S2 (z)) ⋅ (zi − z) = 2|zi |2 − 2|z|2 − 4z ⋅ (zi − z),

(10.32)

10.6 Symmetric transport and congruent multi-partitions | 143

Figure 10.1: Interpolation: z is the mid-point between x and y = T (x).

where we used 21 (S1 (z) + S2 (z)) = z (Figure 10.1). Then, (10.30) and the above imply ∫ {|S1 (z) − zi |2 − |S1 (z) − z|2 + |S2 (z) − zi |2 − |S2 (z) − zi |2 } dλi = 4 ∫ |z|2 dλi − 2λi (ℝd )|zi |2 = 4 ∫ |zi − z|2 dλi . Together with (10.31), (10.32), and (10.8) this implies ϕm (μ, ν) ≤ 4 ∑ ∫ |z − zi |2 dλi 1≤i≤m

(10.33)

for any weak partition λ1 , . . . , λm . Taking the minimal weak partition it turns out by Corollary 10.5.2 that the right side of (10.33) is as small as 4Cd,2 (∫ f d/(d+2) dx)(d+2)/d , where f is the density of λ.

10.6 Symmetric transport and congruent multi-partitions The optimal transport between two (ℝJ+ )-valued measures, discussed in Part II, can be naturally generalized to an optimal transport between two general vector-valued measures. Here we replace the measures μ, ν by ℝJ+ -valued measures μ̄ := (μ(1) , . . . , μ(J) ) ∈ ℳJ+ (X),

ν̄ := (ν(1) , . . . , ν(J) ) ∈ ℳJ+ (Y),

and we denote μ = |μ|̄ := ∑J1 μ(j) , ν = |ν|̄ := ∑J1 ν(j) . The set Π(μ, ν) (9.3) is generalized into Π(μ,̄ ν)̄ := {π ∈ ℳ+ (X × Y); ∫ X

dμ(j) (x)π(dxdy) = ν(j) (dy), j = 1 . . . J}, dμ

(10.34)

where dμi /dμ, dνi /dν stands for the Radon–Nikodym derivative. In general the set Π(μ,̄ ν)̄ can be an empty one. If Π(μ,̄ ν)̄ ≠ 0, then μ̄ ≻ ν̄ (Definition 5.3.2). The generalization of the Kantorovich problem (9.1) takes the following form:

144 | 10 Interpolated costs

θ(μ,̄ ν)̄ := max ∫ ∫ θ(x, y)π(dxdy), π∈Π(μ,̄ ν)̄

θ(μ,̄ ν)̄ = ∞ if μ̄ ⊁ ν.̄

(10.35)

X Y

If J > 1, then θ(μ,̄ ν)̄ ≠ θ(ν,̄ μ)̄ in general, even if μ̄ and ν̄ are living on the same domain X and θ(x, y) = θ(y, x) for any x, y ∈ X. Indeed we obtain from (10.35) that θ(ν,̄ μ)̄ = ∞ if ν̄ ⊁ μ,̄ while θ(μ,̄ ν)̄ < ∞ if μ̄ ≻ ν.̄ This is in contrast to the case J = 1. Let Z be a measure space and let θ satisfy (10.1). Then we define ̄ θ(̄ μ,̄ ν)̄ := sup θ1 (μ,̄ λ)̄ + θ2 (ν,̄ λ).

(10.36)

̄ μ∧ ̄ ν̄ λ≺

In particular, we have the following. ̄ ) = ν(X ̄ ). If X = Y , θ1 = θ2 , then θ(̄ μ,̄ ν)̄ = θ(̄ ν,̄ μ)̄ for any μ,̄ ν.̄ θ(̄ ν,̄ μ)̄ < ∞ iff μ(X ̄ ν) < ∞ only if μ(X ) = ν(Y ). If J = 1, then θ(μ,

From now on we assume that Z is a finite space. One of the motivations for this model is an extension of the hedonic market (Section 10.4) to several commodities. Consider a market of 𝒥 = (1, . . . , J) goods. The domain X is the set of consumers of these goods, and μ(j) is the distribution of consumers of j ∈ 𝒥 . Likewise, Y is the set of manufacturers of the goods, and ν(j) is the distribution of the manufacturers of j ∈ 𝒥 . In addition we presume the existence of N “commodity centers” ZN := {z1 , . . . , zN }. Let θ1 (x, zi ) be the utility for consumer x of good j at center zi , and the same for (j) θ2 (y, zi ) for producer y of good j at center zi . We may extend Definition 10.3.1 of congruent N-partition to this setting: Partitions A⃗ = (A1 , . . . , AN ) of X and B⃗ = (B1 , . . . , BN ) of Y are congruent with respect to μ̄ = (μ(1) , . . . , μ(J) ), ν̄ = (ν(1) , . . . , ν(J) ) if (j)

μ(j) (Ai ) = ν(j) (Bi ),

1 ≤ i ≤ N, 1 ≤ j ≤ J.

(10.37)

Any such possible congruent partition represents a possible matching between the consumers and the producers: all consumers in Ai and all producers in Bi are associated with the single center zi . The balance condition (10.37) guarantees that center zi can satisfy the supply and demand for all goods J simultaneously. The total utility of such a congruent partition is J

Θ({Ai }, {Bi }) := ∑ ∑ ∫ θ1 (x, z)μ(j) (dx) + ∫ θ2 (y, z)ν(j) (dy) j=1 z∈ZN A

(j)

i

(j)

Bi

≡ ∑ ∫ θ1 (x, z)μ(dx) + ∫ θ2 (y, z)ν(dy), z∈ZN A

i

Bi

(10.38)

10.6 Symmetric transport and congruent multi-partitions | 145

where J

μ := ∑ μ(j) , j=1

J

ν := ∑ νj , j=1

θ1 := ∑ θ1 dμ(j) /dμ, (j)

j

θ2 := ∑ θ2 dν(j) /dν. (j)

j

The efficient partition is the one which maximizes the total utility among all possible congruent partitions. Other motivation concerns an application of Monge metric to colored images. The Monge metric (often called the “earth movers metric”) has become very popular in computer imaging in recent years. The general practice for black and white images is to consider these images as probability measures on a Euclidean domain (say, a rectangle B), demonstrating the level of gray. The matching between the two images is reduced to solving the Monge problem for the two corresponding measures μ, ν on B, and is given by the optimal matching T : B → B in (1.15), where, in general, θ(x, y) = −|x − y|2 . The motivation is either to quantify the difference between two such images, or to interpolate between the two images in order to obtain a video connecting two possible states. If these measures are colored, then the general practice is to consider them as probability measures in a lifted space B × C, where the color space C is, in general, a three-dimensional domain representing the level of the RGB (red–green–blue) values. The matching is still given by a solution of the Monge problem (1.15) where, this time, the measures are defined on B×C and the optimal matching is a mapping in this space as well. The alternative paradigm suggested by vectorized transport is to view the images as vector-valued (RGB) measures. It is remarkable, as shown in Lemma 10.1, that the case of a single good (J = 1) is reduced to an optimal transport of (X, μ) to (Y, ν) with respect to the utility θZN (x, y) := max θ1 (x, z) + θ2 (y, z). z∈ZN

This, unfortunately, is not the case for the vectorized case. However, Theorem 10.2 can be extended to the vectorized case, where we define ⃗ ; Ξ(1) (P)⃗ ≡ μ (ξ (1) (⋅, P))

̄ ξ (1) (x, P)⃗ ≡ max(θ1 (x, zi ) + p⃗ i ⋅ dμ/dμ(x)),

⃗ ; Ξ (P)⃗ ≡ ν (ξ (2) (⋅, P))

ξ

(2)

i

(2)

̄ (x, P)⃗ ≡ max(θ2 (y, zi ) + p⃗ i ⋅ dν/dν(y)), i

⃗ Ξ(P)⃗ := Ξ(1) (P)⃗ + Ξ(2) (−P). The proof of the theorem below is very similar to the proof of Theorem 10.2, so we omit it. Theorem 10.5. If any J ≥ 1 and under Assumption 10.3.1, ̄ max Θ({Ai }, {Bi }) = inf Ξ(P;⃗ μ,̄ ν),

{Ai },{Bi }

P⃗

(10.39)

146 | 10 Interpolated costs (j) where the infimum is over all N × J matrices P⃗ = {pi } and the maximum is over all μ̄ − ν̄ congruent partitions. If a minimizer P⃗ 0 is obtained, then the optimal congruent partitions {A0i }, {B0i } satisfy θ

A0i = Ai 1 (P⃗ 0 ),

θ

B0i = Ai 2 (−P⃗ 0 ),

̄ ̄ where Aθi (P)⃗ are defined as in (7.35), where ζ ̄ = dμ/dμ (resp. ζ ̄ = dν/dν).

|

Part IV: Cooperative and non-cooperative partitions

11 Back to Monge: individual values You don’t get paid for the hour. You get paid for the value you bring to the hour. (Jim Rohn)

Theorems 7.8 and 7.9 are the most general result we obtained so far, regarding the existence and uniqueness of generalized, strong (sub)partitions. In particular, it provides a full answer to the questions raised in Section 4.7, together with a constructive algorithm via a minimization of a convex function for finding the optimal (sub)partitions. What we need are just Assumptions 6.2.1 and 7.2.1(i), (ii). Yet, it seems that we still cannot answer any of these questions regarding the saturation and oversaturation cases for non-generalized (sub)partitions, discussed in Sections 4.2–4.4. Let us elaborate this point. Theorem 7.8 provides us with uniqueness only up to a coalition ensemble. So, if the ensemble’s units are not singletons, the theorem only gives us uniqueness up to the given ensemble. On the other hand, Theorem 7.9 (as well as Theorem 7.10) provides uniqueness without reference to any coalition. However, the assumption behind this theorem requires the fixed exchange ratios {zi⃗ } defined in Section 6.2.2 and the corresponding Assumption 6.2.2. The Monge partition problem, as described in Chapter 4, corresponds to the case where ζ ̄ is real-valued (i. e., J = 1). This, indeed, is equivalent to the case of fixed exchange rates in ℝJ , J > 1, where all zi⃗ ∈ ℝJ equal each other. This, evidently, defies Assumption 6.2.2. So, what about Theorem 7.7? It only requires Assumption 7.2.1, which, under the choice ζ ̄ ≡ 1, takes the following form. Assumption 11.0.1. i) For any i, j ∈ ℐ and any r ∈ ℝ, μ (x ∈ X; θi (x) − θj (x) = r) = 0. ii) For any i ∈ ℐ and any r ∈ ℝ, μ (x ∈ X; θi (x) = r) = 0. Hence, Theorem 7.7 can be applied for non-generalized (sub)partitions, granting Assumption 11.0.1. However, this theorem only guarantees the existence and uniquē ness of a strong (sub)partition for interior points of ΔN (μ). Which of the points in ΔN (μ)̄ are interior points? It is evident that under the choice ζ ̄ = 1 the US, S, OS conditions (5.9), (5.8), (5.7) are reduced to (4.7), (4.6), (4.5). Hence, an interior point must be a US point (4.7). In particular, we still cannot deduce the uniqueness of stable partitions for (S) and (OS) capacities... But, alas, “Despair is the conclusion of fools.”1 It turns out that we can still prove this result, using only Assumption 11.0.1(i). We recall the setting of the Monge problem (Chapter 4). Here J = 1, so we set 𝕄+ (N, J) = 𝕄󸀠 (N, J) := ℝN and ζ ̄ = 1. In addition we make the following change 1 Benjamin Disraeli, The Wondrous Tale of Alroy, Part 10, Chapter 17. https://doi.org/10.1515/9783110635485-011

150 | 11 Back to Monge: individual values of notation from Chapters 5–7.2: we replace p⃗ by p⃗ = −p.⃗ This notation is more natural if we interpret p⃗ as the price vector of the agents. Under this change p⃗ 󳨃→ Ξθ (p)⃗ := μ (max(θi (⋅) − pi )) , i∈ℐ

(11.1)

and for Ξθ,+ (equation (4.17)), p⃗ 󳨃→ Ξθ,+ (p)⃗ := μ (max[θi (⋅) − pi ]+ ) . i∈ℐ

(11.2)

Recall N = |ℐ | is the number of agents in ℐ . Let m⃗ = (m1 , . . . , mN ) ∈ ℝN+ and |m|⃗ := In that case the definitions of ΔN and ΔN (Definition 5.1.2) are reduced to

∑Ni=1 mi .

ΔN (μ) := {m⃗ ∈ ℝN+ ; |m|⃗ = μ(X)},

ΔN (μ) := {m⃗ ∈ ℝN+ ; |m|⃗ ≤ μ(X)}.

(11.3)

Theorem 11.1. a) Let Assumption 11.0.1(i) hold. Let K ⊂ ℝN+ be a closed convex set such that |m|⃗ ≥ μ(X) for any m⃗ ∈ K. Then there exists an equilibrium price vector p⃗ 0 , unique up to an additive translation p0i → p0i + γ,

i ∈ ℐ , γ ∈ ℝ,

(11.4)

which is a minimizer of p⃗ 󳨃→ Ξθ (p)⃗ + HK (p)⃗ on ℝN (recall (7.53)). Moreover, the associated partition A⃗ θ (p⃗ 0 ) := (Aθ1 (p⃗ 0 ), . . . , AθN (p⃗ 0 )), where Aθi (p⃗ 0 ) := {x ∈ X; θi (x) − p0i > max θj (x) − p0j }, j=i̸

is the unique optimal partition which maximizes θ(A)⃗ on 𝒪𝒫 NK∩ΔN (μ) .

(11.5)

b) Let Assumption 11.0.1(i), (ii) hold. Let K ⊂ ℝN+ be a closed convex set such that K ∩ ΔN (μ) ≠ 0. Then there exists an equilibrium price vector p⃗ 0 which is a minimizer of p⃗ 󳨃→ Ξ+θ (p)⃗ + HK (p)⃗ on ℝN . Moreover, the associated (sub)partition 0 θ,+ 0 A⃗ θ,+ (p⃗ 0 ) := (Aθ,+ 1 (p⃗ ), . . . , AN (p⃗ )),

11 Back to Monge: individual values | 151

where θ 0 θ 0 θ 0 0 ⃗0 ⃗ ⃗ ⃗ Aθ,+ i (p ) := Ai (p ) − A0 (p ), A0 (p ) := {x ∈ X; max θj (x) − pj ≤ 0},

(11.6)

1≤j≤N

is the unique optimal subpartition which maximizes θ(A)⃗ on 𝒪𝒮𝒫 NK∩Δ

N (μ)

μ(Aθ0 (p⃗ 0 ))

. If

> 0, then the vector p⃗ 0 is unique, and if μ(Aθ0 (p⃗ 0 )) = 0, then p⃗ 0 is unique up to a negative additive translation p0i → p0i − γ,

i ∈ ℐ , γ ∈ ℝ+ .

(11.7)

In particular, recalling Section 7.1.3 we obtain, in spite of the unboundedness of the equilibrium price p⃗ 0 (11.4), the following corollary. Corollary 11.0.1. There is no escalation for the Monge problem under Assumption 11.0.1. Another conclusion which we obtain yields a unified representation in the US, S, ⃗ so HK (p)⃗ = [p]⃗ + ⋅ m,⃗ where and OS cases. Here we consider K = {s⃗ ∈ ℝN+ ; s⃗ ≤ m}, [p]⃗ + := ([p1 ]+ , . . . , [pN ]+ ). Corollary 11.0.2. Under Assumption 11.0.1, there exists a (sub)partition A⃗ 0 such that ⃗ := min Ξθ,+ (p)⃗ + [p]⃗ + ⋅ m⃗ = Ξθ,+ (p⃗ 0 ) + [p⃗ 0 ]+ ⋅ m.⃗ θ(A⃗ 0 ) = Σθ,+ (m) ⃗ I p∈ℝ

Moreover, A⃗ 0 = Aθ,+ (p⃗ 0 ) is given by (11.6). i The claim below is an extension, for Monge (sub)partitions, of Corollary 7.2.1 which uses the uniqueness result of the equilibrium vector p⃗ 0 and Proposition A.10. Corollary 11.0.3. Under Assumption 11.0.1, the function Σθ,+ is differentiable at any interior point m⃗ ∈ ΔN (μ), and 𝜕Σθ,+ = p0i , 𝜕mi

i ∈ ℐ.

If m⃗ ∈ ΔN (μ), then Σθ,+ is differentiable in the “negative” direction, i. e., 𝜕− Σθ,+ ⃗ = p0i , := − lim ϵ−1 (Σθ,(+) (m⃗ − ϵe⃗i ) − Σθ,(+) (m)) ϵ↓0 𝜕mi

(11.8)

while Σθ is differentiable on the tangent space of ΔN (μ), i. e., ⃗ = ζ ̄ ⋅ p⃗ 0 lim ϵ−1 (Σθ,(+) (m⃗ + ϵζ ⃗ ) − Σθ,(+) (m)) ϵ↓0

for any ζ ̄ = (ζ1 , . . . , ζN ) satisfying ∑i∈ℐ ζi = 0, ζi ≥ 0 if mi = 0.

(11.9)

152 | 11 Back to Monge: individual values Remark 11.0.1. The vector p⃗ 0 defined in the saturated case by (11.8) is the maximal price vector (11.7). It is the maximal price which the agents can charge such that any consumer will attend some agent. Remark 11.0.2. The two parts of the theorem contain the three cases (recall (4.5), (4.6), (4.7): ⃗ US) the undersaturated case, m⃗ ∈ int(ΔN (μ)) in part (b) where K = {m}, ⃗ and S) the saturated m⃗ in both (a) and (b) where m⃗ ∈ ΔN (μ), K = {m}, OS) the oversaturated case, where m⃗ ∈ ̸ ΔN (μ). If the components θi are all nonnegative, then case (a) is valid since the only maximizer of Σθ,+ is in ΔN (μ) (show it!). Proof of Theorem 11.1. (a) Inequality (4.18) of Proposition 4.3 is valid also if we replace Ξθ,+ by Ξθ . Indeed, (4.19) is extended to2 θi (x) ≤ ξ (p,⃗ x) + pi ,

where ξ (p,⃗ x) := max θj (x) − pj , 1≤j≤N

so θ(A)⃗ ≤ Ξθ (p)⃗ + HK (p)⃗

(11.10)

holds for any A⃗ ∈ 𝒪𝒮𝒫 NK and p⃗ ∈ ℝN . In the case of equality in (11.10), Proposition 4.6 is valid as well. ⃗ where m⃗ is a saturated vector (m⃗ ∈ ΔN (μ)). ̄ Then (4.18) takes Assume first K = {m}, the form θ(A)⃗ ≤ Ξθ (p)⃗ + p⃗ ⋅ m.⃗

(11.11)

Note that Proposition 7.1 can be applied since Assumption 7.2.1(i) is compatible with Assumption 11.0.1. In particular it follows that Ξθ is differentiable on ℝN . The first equality in (7.42) is translated into 𝜕Ξθ (p)⃗ = − ∫ dμ. 𝜕pi

(11.12)

Aθi (p)⃗

We now prove the existence of such a minimizer p⃗ 0 . Observe that Ξθ (p⃗ + α1)⃗ = Ξθ (p)⃗ − αμ(X);

1⃗ := (1, . . . , 1) ∈ ℝN .

(11.13)

2 Note the change of notation from p⃗ to −p⃗ between Section 4.4 and here. This is because p⃗ is more natural as a price vector in Section 4.4.

11 Back to Monge: individual values | 153

In particular, ∇Ξθ (p)⃗ = ∇Ξθ (p⃗ + α1)⃗ and, in the saturated case 1⃗ ⋅ m⃗ = μ(X): ⃗ Ξθ (p)⃗ + p⃗ ⋅ m⃗ = Ξθ (p⃗ + α1)⃗ + (p⃗ + α1⃗ ⋅ m)

(11.14)

for any α ∈ ℝ. So, we restrict the domain of Ξθ to ℝN0 := {p⃗ ∈ ℝN , p⃗ ⋅ 1⃗ = 0}.

(11.15)

Let p⃗ n be a minimizing sequence of p⃗ 󳨃→ Ξθ (p)⃗ − p⃗ ⋅ m⃗ in ℝN0 , that is, lim Ξθ (p⃗ n ) + p⃗ n ⋅ m⃗ = inf Ξθ (p⃗ n ) + p⃗ n ⋅ m.⃗

n→∞

⃗ N p∈ℝ

Let ‖p‖⃗ 2 := (∑i∈ℐ p2i )1/2 be the Euclidean norm of p.⃗ If we prove that for any minimizing sequence p⃗ n the norms ‖p⃗ n ‖2 are uniformly bounded, then there exists a converging subsequence whose limit is the minimizer p⃗ 0 . This follows since Ξθ is, in particular, a continuous function. ̂⃗ := p⃗ /‖p⃗ ‖ . Assume there exists a subsequence along which ‖p⃗ n ‖2 → ∞. Let p n n n 2 Then ⃗ Ξθ (p⃗ n ) + p⃗ n ⋅ m⃗ := [Ξθ (p⃗ n ) − p⃗ n ⋅ ∇p⃗ Ξθ (p⃗ n )] + p⃗ n ⋅ (∇p⃗ Ξθ (p⃗ n ) + m)

̂⃗ ⋅ (∇ Ξθ (p⃗ ) + m) ⃗ . = [Ξθ (p⃗ n ) − p⃗ n ⋅ ∇p⃗ Ξθ (p⃗ n )] + ‖p⃗ n ‖2 p n p⃗ n

(11.16)

Note that Ξθ (p)⃗ − p⃗ ⋅ ∇Ξθ (p)⃗ = ∑ ∫ θi dμ, i∈ℐ

(11.17)

Ai (p)⃗

so, in particular, ⃗ = ∑ ∫ θi (x)dμ ≤ ∫ max θi dμ < ∞. 0 ≤ ∫ min θi dμ ≤ [Ξθ (p)⃗ − p⃗ ⋅ ∇p⃗ Ξθ (p)] X

i∈ℐ

i∈ℐ

X

Ai (p)⃗

i∈ℐ

(11.18)

By (11.16)–(11.18) we obtain, for ‖p⃗ n ‖2 → ∞, ̂ ⋅ (∇ Ξθ (p⃗ ) + m) ⃗ = 0. p⃗ n

lim p⃗ n→∞ n

(11.19)

̂⃗ lives in the unit sphere in ℝN (which is a compact set), there exists a subseSince p n ̂⃗ → p ̂⃗ := (p̂ , . . . , p̂ ). Let P := max p ̂ 0,i = quence for which p n 0 0,1 0,N + i∈ℐ ̂ i,0 and J+ := {i; p P+ }. Note that for n → ∞ along such a subsequence, pn,i − pn,k → ∞ iff i ∈ J+ , k ∈ ̸ J+ . It follows that Aθk (p⃗ n ) = 0 if k ∈ ̸ J+ for n large enough, and hence μ(⋃i∈J+ Aθi (p⃗ n )) = μ(X) = μ(X) for n large enough. Let μni be the restriction of μ to Aθi (p⃗ n ). Then the limit μni ⇀ μi exists (along a subsequence) where n → ∞. In particular, by (11.12) 𝜕Ξθ (p⃗ n ) = − lim ∫ dμni = − ∫ dμi , n→∞ n→∞ 𝜕p n,i lim

X

X

154 | 11 Back to Monge: individual values ̂ 0,i = P+ for i ∈ J+ is the maximal while μi ≠ 0 only if i ∈ J+ , and ∑i∈J+ μi = μ. Since p ̂ value of the coordinates of p⃗ , it follows that 0

̂⃗ ⋅ m⃗ − P μ(X). ̂⃗ ⋅ m⃗ − P ∑ ∫ dμ = p ̂ ⋅ (∇ Ξθ (p⃗ ) + m) ⃗ =p 0 + i 0 + p⃗ n

lim p⃗ n→∞ n

i∈J+ X

̂⃗ ⋅ m⃗ < P μ(X) unless J = {1, . . . , N}. In the last case we obtain Now, by definition, p 0 + + ̂⃗ is in the unit ̂⃗ = 0, which contradicts p a contradiction of (11.15) since it implies p 0 0 N sphere in ℝ . If J+ is a proper subset of {1, . . . , N} we obtain a contradiction to (11.19). Hence ‖p⃗ n ‖2 is uniformly bounded, and any limit p⃗ 0 of this set is a minimizer. The proof of uniqueness of optimal partition is identical to the proof of this part in Theorem 7.7 (equation (7.44)). This also implies the uniqueness (up to a shift) of p⃗ 0 via (11.14). To complete the proof we need to show that ⃗ := min Ξθ (p)⃗ + p⃗ ⋅ m⃗ m⃗ ∈ K ∩ ΔN (μ) 󳨃→ Σθ (m) ⃗ N p∈ℝ

(11.20)

admits a unique maximizer. Recall that the function Ξθ is a convex function on ℝN . Moreover, its partial derivatives exist at any point in ℝN , which implies that its subgradient is a singleton. Its Legendre transform takes finite values only on the simplex of saturated vectors ΔN (μ). Indeed, by (11.13) Ξθ (p⃗ + α1)⃗ + (p⃗ + α1)⃗ ⋅ m⃗ = Ξθ (p)⃗ + α(∑ mi − μ(X)), i∈ℐ

so ⃗ := sup p⃗ ⋅ m⃗ − Ξθ (p)⃗ = ∞ Ξθ,∗ (m) ⃗ N p∈ℝ

if m⃗ ∈ ̸ ΔN (μ). In fact, we already know that ΔN (μ) is the essential domain of Ξθ,∗ . Now, K ∩ ΔN (μ) is a compact, convex set. The uniqueness of the maximizer (11.20) follows if Ξθ,∗ is strictly convex on its essential domain ΔN (μ). This follows from the differentiability of Ξθ and from Proposition A.10. b) The proof of case (b) follows directly from the proof of case (a), where we add the agent {0} to {1, . . . , N}, and set θ0 ≡ 0. The uniqueness of p⃗ 0 = (p01 , . . . , p0N ) in that case follows from the uniqueness up to a shift of (p00 , p01 , . . . , p0N ), where we “nailed” this shift by letting p00 = 0.

11.1 The individual surplus values The main conclusion we may draw from Theorem 11.1 is the existence of an “individual value” (i. v.) for an agent. This is the value which the consumers attribute to their

11.1 The individual surplus values | 155

agents. If the price vector of agents is p,⃗ then the individual value for agent i is Viθ (p)⃗ := ∫ θi dμ,

(11.21)

Aθi (p)⃗

where Aθi (p)⃗ = {x; θi (x) − pi = maxj=i̸ [θj (x) − pj ]+ }. Under the conditions of Theorem 11.1 we know that the partition is uniquely determined by the capacities m,⃗ so we may consider the partition A⃗ and the individual values V⃗ as functions of the capacity vector m,⃗ rather than the price vector p.⃗ Thus, we sometimes refer to ⃗ := ∫ θi dμ, Vi (m) ⃗ Aθi (m)

⃗ = Aθi (p(⃗ m)). ⃗ where Aθi (m) Example 11.1.1 (The case of a single agent). For θ ∈ C(X) being the utility function of a single agent, let (Figure 11.1) Aθp := {x ∈ X; θ(x) ≥ p}, mθ (p) := μ(Aθp ),

∞

Fθ (t) := ∫ mθ (s)ds, t

ℱθ (m) := inf[mt + Fθ (t)]. t∈ℝ

Note that Fθ is defined since θ is bounded on X, so mθ (t) = 0 for t > maxX θ. Moreover, Fθ and ℱθ are concave functions, and ∞

− ∫ tdmθ (t) = ∫ θdμ ≡ V θ (mθ (p)). p

Aθp

Integration by parts and duality implies ∞

− ∫ tdmθ (t) = pmθ (p) + Fθ (p) ≡ ℱθ (mθ (p)). p

Figure 11.1: Single agent.

156 | 11 Back to Monge: individual values Substituting mθ (p) = m we obtain that the i. v. for the single agent of capacity m is just ℱθ (m), so, for any m > 0, V θ (m) = ℱθ (m)

∀m ∈ (0, μ(X)].

(11.22)

Note that ℱθ (m) = −∞ if m > μ(X). The equilibrium price p = pθ (m) corresponding to capacity m is the inverse of the function mθ (p), and by duality pθ (m) =

dℱθ (m) . dm

(11.23)

Also, by definition, ℱθ (0) = 0, so lim m−1 V θ (m) =

m→0

d ℱ (0) = max θ, X dm θ

as expected. Example 11.1.2 (The marginal case of two agents undersaturation). Assume N = 2 and m1 + m2 = μ(X). Using the notation of Example 11.1.1 we consider (Figure 11.2) Aθp1 −θ2 := {x ∈ X; θ1 (x) ≥ θ2 (x) + p} .

(11.24)

θ −θ

The complement of this set is, evidently, A−p2 1 . Since A+0 = 0 in the saturated case, we obtain by Theorem 11.1(a) that the equilibrium price is determined by any (p1 , p2 ) θ −θ such that p = p2 − p1 verifies μ(Ap1 2 ) = m1 . Since m2 = μ(X) − m1 , it implies that θ −θ

μ(A−p2 1 ) = m2 as well.

Figure 11.2: Two agents in saturation.

However, the i. v. is not given by (11.22) as in Example 11.1.1. In particular, in the limit m1 → 0,

11.2 Will wiser experts always get higher values?

lim

m1 →0

| 157

V1 (m1 ) ∈ Conv({θ1 (x1 )}), m1

where {x1 } is the set of maximizers of θ1 − θ2 and Conv(⋅) is the convex hull of this set in ℝ1 .

Example 11.1.3. Suppose θ is a non-negative, continuous function on X verifying μ(x; θ(x) = r) = 0 for any r ≥ 0. Let λN > λN−1 > ⋅ ⋅ ⋅ λ1 > 0 be constants. We assume that θi := λi θ, where μ(x; θ(x) = r) = 0 ∀r ∈ ℝ (in particular, θ⃗ = (θ1 , . . . , θN ) verifies Assumption 11.0.1). Let m⃗ ∈ ΔN (1). From (4.14), (4.15) we obtain A+0 (p)⃗ ≡ {x; θ(x) < min λi−1 pi }, A+i (p)⃗

≡ {x; min j>i

pj − pi λj − λi

i∈ℐ

> θ(x) > max

pj − pi

j 0 is a constant. Case 2: θ̃1 = βθ1 , where β > 2 is a constant. In the first case the “gaps” θ1 −θ2 and θ̃1 −θ2 preserve their order, so if x1 is a maximizer of the first, it is also a maximizer of the second. In particular the optimal partition is unchanged, and we can even predict that Ṽ 1 = V1 + λm1 > V1 (Theorem 11.2 below). In the second case the order of gaps may change. It is certainly possible that ̃θ (x̃ ) − θ (x̃ ) > θ̃ (x ) − θ (x ) (where x , x̃ are as above), but if this is the case, an 1 1 2 1 1 1 2 1 1 1 elementary calculation yields θ̃1 (x̃1 ) > θ1 (x1 ), so the above argument fails. Indeed, if we assume both βθ1 (x̃1 ) − θ2 (x̃1 ) > βθ1 (x1 ) − θ2 (x1 ) and βθ1 (x̃1 ) < θ1 (x1 ), then (since

11.2 Will wiser experts always get higher values?

| 159

β ≥ 2) θ1 (x1 ) − θ2 (x1 ) < −θ2 (x̃1 ) < θ1 (x̃1 ) − θ2 (x̃1 ), so x1 cannot be the maximizer of θ1 − θ2 , as assumed. In fact, we can get the same result if either θ̃1 ≥ 2θ1 or θ̃1 = βθ1 and β ≥ 1 (but, remarkably, not in the case θ̃1 ≥ βθ1 where β < 2!). This follows from the following results. ̃ ̃ Theorem 11.2 ([51]). Let θ⃗ := (θ1 , . . . , θN ) and θ⃗ := (θ̃1 , θ2 , . . . , θN ). Assume both θ,⃗ θ⃗ verN ify Assumption 11.0.1. Let m⃗ ∈ ℝ+ , V1 is the i. v. of agent 1 corresponding to θ⃗ and the ̃ capacity m,⃗ and Ṽ 1 is the same corresponding to θ⃗ and the same capacity m.⃗ i) If θ̃1 = βθ1 for a constant β > 0, then Ṽ 1 ≥ βV1 if β > 1 and Ṽ 1 ≤ βV1 if β < 1. ii) If m⃗ is either saturated or undersaturated and θ̃1 = θ1 + λ for a constant λ > 0, then Ṽ 1 = V1 + λm1 . In Theorem 11.3 we expand on case (i) of Theorem 11.2 and obtain the following somewhat surprising result. Theorem 11.3. Under the same conditions as Theorem 11.2: i) Suppose θ̃1 ≥ βθ1 , where β > 1 is a constant. Then Ṽ 1 ≥ (β − 1)V1 .

(11.28)

̃ ⃗ and (θ,⃗ m), ⃗ where m⃗ is a saturation ii) For any β > 2, s > β−1 there exists a system (θ,⃗ m) ̃ vector, such that θ̃ ≥ βθ , θ̃ = θ for i ≠ 1, both θ,⃗ θ⃗ verify Assumption 11.0.1, and 1

1

i

i

Ṽ 1 < sV1 . In particular, inequality (11.28) is sharp in the case β > 2. Corollary 11.2.1. The i. v. of an agent cannot decrease if its utility θi is replaced by θ̃i ≥ 2θi , without changing any of the capacities and the utilities of other agents. In Theorem 11.4 we obtain sharp conditions for the decrease of i. v., given an increase of the corresponding utility. Theorem 11.4. Under the assumption of Theorem 11.2, if m⃗ is either undersaturated or saturated: i) If 1 < β < 2, λ ≥ 0, and βθ1 (x) ≤ θ̃1 (x) ≤ βθ1 (x) + λ,

(11.29)

m λ(2 − β) Ṽ 1 ≥ V1 − 1 . β−1

(11.30)

then

160 | 11 Back to Monge: individual values ̃ ⃗ and (θ,⃗ m) ⃗ ii) For any 1 < β < 2, λ > 0, s < (2 − β)/(β − 1) there exists a system (θ,⃗ m) ̃ ̃ ̃ ⃗ ⃗ such that βθ ≤ θ ≤ βθ + λ, θ = θ for i ≠ 1 and both θ, θ verify Assumption 11.0.1 such that

1

1

1

i

i

Ṽ 1 < V1 − m1 λs. In particular, inequality (11.30) is sharp in the case 1 < β < 2. 11.2.1 Proofs The key lemma is an adaptation of Lemma 7.1. ⃗ t) : X × [0, a] → ℝN for any t ∈ [0, a]. Let θ⃗ verify Lemma 11.1. Let a > 0 and θ⃗ := θ(x, + Assumption 11.0.1 for t = 0 and t = a. Assume further that each component t 󳨃→ θi (x, t) is convex and differentiable on [0, a] for any x ∈ ℝN and 𝜕t θi := θi̇ ∈ 𝕃∞ (X × [0, a]) for any i ∈ ℐ . Then the function (p,⃗ t) 󳨃→ Ξθ(⋅,t) (p)⃗ (11.1) is convex on ℝN × [0, a], and if its ⃗ t derivative Ξ̇ θ(⋅,t) (p) exists at (p,⃗ t), then ⃗ Ξ̇ θ(⋅,t) (p) = ∑ ∫ θ̇i (x, t)dμ. ⃗

(11.31)

Ai (p,⃗ t) := {x ∈ X; θi (x, t) − pi > θj (x, t) − pj ∀j ≠ i}.

(11.32)

i∈ℐ

⃗ Ai (p,t)

Here

The same holds if we replace Ξθ by Ξθ,+ ((11.2) and (11.32)) by A+i (p,⃗ t) := {x ∈ X; θi (x, t) − pi > [θj (x, t) − pj ]+ ∀j ≠ i}.

(11.33)

Proof. The proof follows as in Lemma 7.1. Here we define ξ : X × ℝN × [0, a] → ℝ, ξ (x, p,⃗ t) = max[θi (x, t) − pi ] i∈ℐ

and Ξθ (p,⃗ t) := ∫X ξ (x, p,⃗ t)μ(dx). Again, ξ is convex on ℝN × [0, a] for any x ∈ X so Ξθ is convex on ℝN × [0, a] as well, while ξ̇ = { implies (11.31).

̇ t) θ(x, 0

if x ∈ Ai (p,⃗ t), if ∃j ≠ i, x ∈ Aj (p,⃗ t)

11.2 Will wiser experts always get higher values?

| 161

Proof of Theorem 11.2. i) Let t ⃗ := (t1 , . . . , tN ) ∈ ℝN . Let Θ(x, t) := (t1 θ1 (x), . . . , tN θN (x)). Consider Ξ(p,⃗ t)⃗ := ΞΘ .

(11.34)

By Lemma 11.1, (p,⃗ t)⃗ 󳨃→ ΞΘ is mutually convex on ℝN × ℝN , and ⃗ 𝜕ti ΞΘ (p,⃗ t)⃗ = ∫ θi dμ ≡ ti−1 Vi (t),

(11.35)

Ai (p,⃗ t)⃗

where Ai (p,⃗ t)⃗ := {x ∈ X; ti θi (x) − pi < tj θj (x) − pj ∀j ≠ 1},

(11.36)

whenever 𝜕ti ΞΘ exists. It follows that both Σ(m,⃗ t)⃗ := min ΞΘ (p,⃗ t)⃗ + m⃗ ⋅ p⃗ ⃗ N p∈ℝ

(11.37)

in the US and S cases or Σ(m,⃗ t)⃗ := max min ΞΘ (p,⃗ t)⃗ + m⃗ ⋅ p⃗ ⃗ m⃗ p∈ℝ ⃗ N m≤

in the OS case are convex with respect to t ⃗ as well. Then 𝜕ti Σ =

∫

θi dμ ≡ ti−1 Vi (t)⃗

Ai (p⃗ 0 ,t)⃗

holds as well, where p⃗ 0 := p⃗ 0 (m,⃗ t)⃗ is the unique equilibrium price vector (perhaps up to an additive constant) guaranteed by Theorem 11.1 for the utility vector Θ. ⃗ := (β, 1; . . . 1) we obtain Hence, for t(β) ⃗ )/β ≡ 𝜕β Σ(m,⃗ t(β) ⃗ ) ≥ 𝜕t Σ(m,⃗ 1)⃗ ≡ V1 (t(1) ⃗ ), V1 (t(β) 1 ⃗ ) ≡ V1 and V1 (t(β) ⃗ ) ≡ Ṽ 1 by (11.35). where V1 (t(1) ii) If θ1 → θ1 + λ, then the optimal partition in the S and US cases is unchanged. Then Ṽ 1 := ∫ (θ1 + λ)dμ = ∫ θ1 dμ + λ ∫ dμ = V1 + λm1 . A1

A1

A1

Proof of Theorem 11.3. i) Let σ := θ̃1 − βθ1 ≥ 0, α := β − 1 ≥ 0. Let a function ϕ : [0, 1] 󳨃→ ℝ satisfy ̇ ϕ(0) = ϕ(0) =0

and ϕ̈ ≥ 0 for any t ∈ [0, 1], ϕ(1) = 1.

(11.38)

162 | 11 Back to Monge: individual values Define θ(x, t) := (1 + αt)θ1 (x) + σ(x)ϕ(t).

(11.39)

θ(x, 1) = θ̃1 (x)

(11.40)

So

̇ t) = αθ (x)+σ(x)ϕ(t). ̇ and θ is convex in t ∈ [0, 1] for any x. Also θ(x, Let now δ > 0. 1 Then ̇ 1) − δ‖σ‖ θ(x, 1) ≥ θ(x, ∞

(11.41)

̇ σ(x)ϕ(1) ≤ σ(x) + θ1 (x) + δ‖σ‖∞ .

(11.42)

provided

̇ Since θ1 and σ are non-negative, the latter is guaranteed if ϕ(1) ≤ 1 + δ. So, we 1+ϵ choose ϕ(t) := t for some ϵ ∈ (0, δ]. This meets (11.38), (11.42). Let now Σ(m,⃗ t) := inf ΞΘ (p,⃗ t) + p⃗ ⋅ m,⃗ ⃗ N p∈ℝ

where Θ(x, t) := (θ(x, t), θ2 (x), . . . , θN (x)). By Lemma 11.1, (p,⃗ t) 󳨃→ ΞΘ (p,⃗ t) is convex. So Σ is convex in t for a fixed m.⃗ In the OS case Σ(m,⃗ t) := sup inf ΞΘ (p,⃗ t) + p⃗ ⋅ m⃗ ⃗ N ⃗ m⃗ p∈ℝ m≤

is convex (as maximum of convex functions) as well. By the same lemma, ̇ 0)dμ = α ∫ θ dμ ≡ αV , Σ(̇ m,⃗ 0) = ∫ θ(, 1 1 A1 (0)

(11.43)

A1 (0)

where A1 (0) is the first component in the optimal partition associated with θ,⃗ while at t = 1 we obtain from convexity and (11.41) ̇ 1)dμ ≤ ∫ (θ(x, 1) + δ‖σ‖ )dμ ≤ ∫ θ(x, 1)dμ + δμ(X)‖σ‖ , Σ(̇ m,⃗ 1) = ∫ θ(x, ∞ ∞ A1 (1)

A1 (1)

A1 (1)

(11.44)

where A1 (1) is the first component in the optimal partition associated with Θ at t = 1. Since τ 󳨃→ ϕ(τ) is convex, τ 󳨃→ Σ(m,⃗ τ) is convex as well by Lemma 11.1 and we get Σ(̇ m,⃗ 1) ≥ Σ(̇ m,⃗ 0).

(11.45)

11.2 Will wiser experts always get higher values?

| 163

From (11.44), (11.45) ∫ θ(x, 1)dμ ≥ αV1 − δμ(X)‖σ‖∞ . A1 (1)

Now, recall β := 1 + α and θ(x, 1) := θ̃1 by (11.40), so ∫A (1) θ(x, 1)dμ ≡ Ṽ 1 . Since δ > 0 1 is arbitrary small, we obtain the result. ii) Assume N = 2, m1 +m2 = μ(X). We show the existence of non-negative, continuous θ1 , θ2 , x1 , x2 ∈ X and λ > 0 such that, for given δ > 0, a) Δ(x) := θ1 (x) − θ2 (x) < Δ(x1 ) for any x ∈ X − {x1 }, b) Δβ (x) := βθ1 (x) − θ2 (x) < Δβ (x1 ) for any x ∈ X − {x1 }, c) Δβ (x2 ) + λ = Δβ (x1 ) + δ. We show that (a)–(c) are consistent with sθ1 (x1 ) > βθ1 (x2 ) + λ

(11.46)

for given s > β − 1. Suppose (11.46) is verified. Let θ0 := {

1−

|x−x2 | ϵ

0

if |x − x2 | ≤ ϵ, if |x − x2 | > ϵ

(11.47)

(assuming, for simplicity, that X is a real interval). Set θ̃1 := βθ1 + λθ0 . If ϵ is small enough, then θ̃1 − θ2 is maximized at x2 by (b), (c), while θ1 − θ2 is maximized at x1 by (a). Letting M1 [Δ (x ) − Δ(x2 )] + λ. β−1 β 1 β−1 β 2 From (c) we obtain β s [Δ (x ) − Δ(x1 )] > [Δ (x ) − Δ(x2 )] + Δβ (x1 ) − Δβ (x2 ) + δ, β−1 β 1 β−1 β 2 that is, (s − β + 1)Δβ (x1 ) − Δβ (x2 ) > (s − β)Δ(x2 ) + s(Δ(x1 ) − Δ(x2 )) + δ(β − 1).

(11.48)

We now set Δβ (x1 ) and λ large enough, keeping δ, Δβ (x2 ), Δ(x1 ), Δ(x2 ) fixed. Evidently, we can do it such that (c) is preserved. Since s − β + 1 > 0 by assumption, we can get (11.48).

164 | 11 Back to Monge: individual values Proof of Theorem 11.4. i) Let β = 1 + t, where t ∈ (0, 1). We change (11.39) into θ(x, t) := (1 + t)(θ1 (x) + γ) + σ(x)ϕ(t)

(11.49)

θ̃1 (x) := (1 + t)θ1 (x) + σ(x)ϕ(t),

(11.50)

and

̇ t) = θ (x) + γ + σ(x)ϕ(t), ̇ where γ > 0 is a constant and σ ≥ 0 on X. Then θ(x, and 1 we obtain ̇ t), t > 0; θ(x, t) ≥ θ(x,

̇ 0) = θ (x) + γ θ(x, 1

(11.51)

provided ̇ ≤ σ(x)ϕ(t) + t(θ (x) + γ); σ(x)ϕ(t) 1

̇ ϕ(0) = 0.

(11.52)

Since θ1 , σ are non-negative, the latter is guaranteed if ̇ ≤ ϕ(t) + ϕ(t)

tγ ̇ ; ϕ(0) = 0. ‖σ‖∞

(11.53)

Since t < 1 (by assumption β := 1 + t < 2), the choice ϕ(τ) := τ1+ϵ for 0 ≤ τ ≤ t and ϵ > 0 small enough (depending on t) verifies (11.53) provided ‖σ‖∞ < γt/(1 − t).

(11.54)

Hence, we can let σ be any function verifying (11.54). Then (11.49), (11.50) imply (1 + t)θ1 (x) ≤ θ̃1 (x) ≤ (1 + t)θ1 (x) +

γt 2+ϵ . 1−t

(11.55)

Now, we note from the second part of (11.51) that ̇ 0)dμ = ∫ (θ + γ)dμ ≡ V + γm Σ(̇ m,⃗ 0) = ∫ θ(, 1 1 1 A1 (0)

(11.56)

A1 (0)

since A1 (0) is independent of γ in the S and US cases. In addition, (11.49), (11.50), (11.53) imply ̇ t)dμ ≤ ∫ θ(⋅, t)dμ = ∫ (θ̃ + (1 + t)γ)dμ Σ(̇ m,⃗ t) = ∫ θ(, 1 A1 (t)

A1 (t)

≡ Ṽ 1 + (1 + t)γm1 ,

A1 (t)

11.2 Will wiser experts always get higher values?

| 165

where A1 (t) is the first component in the optimal partition associated with Θ. Since τ 󳨃→ ϕ(τ) is convex, τ 󳨃→ Σ(m,⃗ τ) is convex as well by Lemma 11.1 and we get, as in (11.45), Σ(̇ m,⃗ t) ≥ Σ(̇ m,⃗ 0),

(11.57)

where, again, we used that A1 (t) is independent of γ and t > 0. Recalling β := 1 + t and letting λ := γ(β − 1)2 /(2 − β) and ϵ small enough, we get (11.29), (11.30), using (11.55), (11.56), (11.57). ii) Assume N = 2, m1 + m2 = 1, that θ1 − θ2 attains its maximum at x1 , and x2 ≠ x1 . Let θ̃1 := βθ1 + λθ0 where θ0 is as defined in (11.47). We assume, as in part (ii) of the proof of Theorem 11.3, that x1 is a maximizer of βθ1 − θ2 as well. Next, assume βθ1 (x1 ) − θ2 (x1 ) < λ + βθ1 (x2 ) − θ2 (x2 ),

(11.58)

which implies, in particular, that x2 is the maximizer of θ̃1 − θ2 (see part (ii) of the proof of Theorem 11.3). If, in addition, θ1 (x1 ) − βθ1 (x2 ) − λ − s > 0,

(11.59)

then, from Example 11.1.2, we obtain the proof for small m1 and V1 ≈ θ1 (x1 )m1 > m1 (θ̃1 (x2 ) + s) ≈ Ṽ 1 + sm1 .

(11.60)

From (11.58) and since x1 is a maximizer of θ1 − θ2 , λ > (β − 1)(θ1 (x1 ) − θ1 (x2 )), so (11.59) and (11.58) are compatible provided λ > (β − 1)2 θ1 (x2 ) + (β − 1) [λ + s] , namely, λ

2−β > (β − 1)θ1 (x2 ) + s. β−1

(11.61)

Thus, if we assume further that, say, θ1 (x2 ) = 0 (which is consistent with the assumption that θ1 , θ2 ≥ 0), then (11.61) is verified for s < λ(2 − β)/(β − 1).

12 Sharing the individual value Share it fairly but don’t take a slice of my pie. (Pink Floyd)

The i. v. of an agent is the surplus she produces for her clients. The question we are going to address is the following: How does an agent share her i. v. with her clients?

We already now that, under a prescribed capacity vector m,⃗ the price that agent i charges for her service is determined by pi . Recall Ξθ,+ (p)⃗ := ∫ max(θi (x) − pi )+ dμ; X

i∈ℐ

⃗ = min Ξθ,+ (p)⃗ + p⃗ ⋅ m.⃗ Σθ (m) ⃗ I p∈ℝ

(12.1)

The relation between the capacity and price is given by pi =

𝜕Σθ , 𝜕mi

mi = −

𝜕Ξθ,+ , 𝜕pi

(12.2)

provided Ξθ,+ and Σθ are differentiable. The profit 𝒫i of agent i fixing a price pi is just pi mi . The residual profit of her consumers is 𝒞i := Vi − 𝒫i , where Vi is the individual value.

Using the duality relation (12.2) we can determine the profit of the agent in terms of either the prices p⃗ charged by the group of agents or the capacity vector m:⃗ 𝒫i (p)⃗ := −pi

𝜕Ξθ,+ , 𝜕pi

⃗ = mi 𝒫i (m)

𝜕Σθ , 𝜕mi

(12.3)

and we use 𝒫i for both representations, whenever no confusion is expected. There is, however, another possibility: In addition to (or instead of) the fixed, flat price pi of her service the agent may charge a commission. This commission is a certain proportion, say, qi ∈ [0, 1), of the gross profit θi (x) she makes for consumer x. In that case, the profit of agent i out of a single consumer x is just pi + qi θi (x), while the net profit of this consumer is (1 − qi )θi (x) − pi . Given a price vector p⃗ = (p1 , . . . , pN ) ∈ ℝN+ and a commission vector q⃗ = (q1 , . . . , qN ) ∈ [0, 1)N , the part of the population not attending any agent is Aθ0 (p,⃗ q)⃗ := {x ∈ X; max(1 − qj )θj (x) − pj ≤ 0}. j

The population attending agent i is, then, θ θ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ Aθ,+ i (p, q) := Ai (p, q) − A0 (p, q), https://doi.org/10.1515/9783110635485-012

168 | 12 Sharing the individual value where Aθi (p,⃗ q)⃗ := {x ∈ X; (1 − qi )θi (x) − pi > max(1 − qj )θj (x) − pj } j=i̸

(12.4)

(compare to (11.5), (11.6)). The profit 𝒫i of agent i fixing a price pi and commission qi is pi mi + qi Vi , where ⃗ , mi (p,⃗ q)⃗ := μ (Aθi (p,⃗ q))

Vi (p,⃗ q)⃗ :=

θi dμ.

∫ Aθi (p,⃗ q)⃗

The residual profit of her consumers is 𝒞i := (1 − qi )Vi − pi mi .

Can we express this profit in terms of “potential functions” as in (12.3)? For this we generalize (12.1) into Ξθ,+ (p,⃗ q)⃗ := ∫ max((1 − qi )θi (x) − pi )+ dμ X

i

and the dual function Σθ (m,⃗ q)⃗ := inf Ξθ,+ (p,⃗ q)⃗ + p⃗ ⋅ m.⃗ ⃗ N p∈ℝ

The condition for differentiability of Ξθ,+ and Σ is the following generalization of Assumption 11.0.1 Assumption 12.0.1. For any i, j ∈ {1, . . . , N} and any r ∈ ℝ, α > 0, μ (x ∈ X; θi (x) − αθj (x) = r) = 0. Under Assumption 12.0.1 we obtain that Ξθ,+ is differentiable in both variables, provided q⃗ ∈ [0, 1)N . Recalling Corollary 11.0.3 we obtain that Σθ is also differentiable with respect to m⃗ for fixed q⃗ ∈ [0, 1)N for any undersaturated m,⃗ (and differentiable ⃗ 1 Moreover, it can be shown that Σθ is also in the negative direction for saturated m). differentiable with respect to q⃗ for any m⃗ in the simplex ΔN (11.3), so the i. v. of agent i is given in either (p,⃗ q)⃗ or (m,⃗ q)⃗ representation as Vi (p,⃗ q)⃗ = −

𝜕Ξθ,+ ; 𝜕qi

Vi (m,⃗ q)⃗ = −

𝜕Σθ . 𝜕qi

− θ ⃗ ⃗ 1 Recall, by Remark 11.0.1, that p(⃗ m,⃗ q)⃗ := −∇m ⃗ Σ (m, q) is, in the saturated case, the maximal price vector charged by the agents.

12.1 Maximizing the agent’s profit |

169

⃗ Thus, we obtain the profit of agent i as a function of either (p,⃗ q)⃗ or (m,⃗ q): 𝒫i (p,⃗ q)⃗ := − (pi

𝜕Ξθ,+ 𝜕Ξθ,+ + qi ), 𝜕pi 𝜕qi

𝒫i (m,⃗ q)⃗ = mi

𝜕Σθ 𝜕Σθ − qi . 𝜕mi 𝜕qi

(12.5)

Note that (12.5) reduces to (12.3) if q⃗ = 0.

12.1 Maximizing the agent’s profit It is, evidently, more natural for an agent to maximize her profit rather than her individual value. Let us consider first the case of a single agent which does not collect a commission. If the utility function for this agent is θ, the flat price she collects is a maximizer of the function p → 𝒫 (p), where 𝒫 (p) = pμ(x; θ(x) ≥ p).

Note that 𝒫 is non-negative for any p ∈ ℝ. Moreover, it is positive in the domain 0 < p < θ̄ := max θ. If (as we assume throughout this book) θ is a bounded continuous function and X is compact, then θ̄ is always obtained in X. However, the maximizer many not be unique. Example 12.1.1. Let (X, μ) = ([0, 1], dx) and let θ be positive on [0, 1), monotone decreasing, and θ(1) = 0. For p ∈ [0, θ(0)] we get m(p) = θ−1 (p), so 𝒫 (p) = pθ−1 (p). Non-uniqueness of max 𝒫 (p) can be visualized easily (Figure 12.1).

Figure 12.1: The two gray rectangles maximize the area of all rectangles below the graph of θ in the positive quadrature whose edges are parallel to the axis.

If we also allow a commission q, then the situation is changed dramatically. Evidently, 𝒫(0, q) can 1 approach the i. v. (= ∫0 [θ(x)]+ dx) arbitrarily close as q ↑ 1.

170 | 12 Sharing the individual value

12.2 Several agents: Nash equilibrium The case of several agents is much more complicated. Let 𝒫i = 𝒫i (p,⃗ q)⃗ be the profit of agents i for given price-commission vectors p,⃗ q.⃗ A natural definition of an equilibrium is the Nash equilibrium. It is satisfied whenever each agent i chooses his strategy (i. e., his price-commission value (pi , qi )) to maximize his profit, assuming that his choice does not affect the choices of other agents. Definition 12.2.1. The vectors p⃗ = (p1 , . . . , pN ) ∈ ℝN , q⃗ = (q1 , . . . , qN ) ∈ [0, 1]N are said to be in Nash equilibrium if 𝒫i (p⃗ −i , pi ; q⃗ −i , qi ) ≤ 𝒫i (p,⃗ q)⃗ 󸀠

󸀠

for any p󸀠i ∈ ℝ, qi󸀠 ∈ [0, 1), and i ∈ ℐ . Here p⃗ −i is the vector p⃗ where the i-coordinate is omitted. The same goes for q⃗ −i . If no commission is charged, the Nash equilibrium p⃗ is defined with respect to flat prices only: ⃗ 𝒫i (p⃗ −i , pi ) ≤ 𝒫i (p), 󸀠

⃗ where 𝒫 (p)⃗ := 𝒫 (p,⃗ 0). An equivalent definition can be given in terms of the dual variable m⃗ (capacities) and q.⃗ In this sense, the agents may control their capacities (instead of the flat prices) and their commissions. Using this, we may assume the existence of capacity constraints m⃗ ≤ m⃗ ∗ and define the constraint Nash equilibrium. Definition 12.2.2. The capacity vector m⃗ ≤ m⃗ ∗ and commission vector q⃗ are said to be in m⃗ ∗ -conditioned Nash equilibrium if 𝒫i (m⃗ −i ; , q⃗ −i , qi ) ≤ 𝒫i (m,⃗ q)⃗ 󸀠

for any m󸀠i ≤ m∗,i , qi󸀠 ∈ [0, 1), and i ∈ {1, . . . , N}. Again, if no commission is charged, the Nash equilibrium m⃗ ≤ m⃗ ∗ conditioned on m⃗ ∗ is defined with respect to the capacities which are determined by the flat prices: ⃗ 𝒫i (m⃗ −i , mi ) ≤ 𝒫i (m), 󸀠

∀m󸀠i ≤ M∗,i ,

⃗ ⃗ := 𝒫 (m,⃗ 0). where 𝒫 (m) If, in addition, the functions 𝒫i are differentiable as well, then we obtain the necessary conditions for a Nash equilibrium. Proposition 12.1. If (p⃗ 0 , q⃗ 0 ) is a Nash equilibrium and 𝒫i are differentiable at p⃗ 0 , q⃗ 0 , then 𝜕𝒫i /𝜕pi = 𝜕𝒫i /𝜕qi = 0 at (p⃗ 0 , q⃗ 0 ).

12.3 Existence of Nash equilibrium

| 171

If (m⃗ 0 , q⃗ 0 ) is an m⃗ ∗ -conditioned Nash equilibrium, then 𝜕𝒫i /𝜕mi ≥ 0,

𝜕𝒫i /𝜕qi = 0

at (m⃗ 0 , q⃗ 0 )

and 𝜕𝒫i /𝜕mi = 0 if m0,i < m∗,i . ⃗ holds if no commission Evidently, the same condition with respect to p⃗ (resp. m) is imposed (q⃗ 0 = 0).

12.3 Existence of Nash equilibrium In general, the existence of Nash equilibrium is not guaranteed. There are, however, some cases in which a conditioned Nash equilibrium exists. For example, if the capacities m⃗ ∗ are sufficiently small, then we expect that, at least if no commission is imposed, the “saturated” capacity m⃗ 0 = m⃗ ∗ is an m⃗ ∗ -conditioned Nash equilibrium. In general, however, there always exists a Nash equilibrium if we allow mixed states [33].

12.4 Efficiency A (sub)partition is called efficient if the sum of all i. v.’s of all agents is maximized. Here we pose no restriction on the capacities. Alternatively, a (sub)partition is efficient if each consumer x attends the agent i which is best for him, provided the utility of this agent is positive. See the following definition. Definition 12.4.1. A (sub)partition (A1 , . . . , AN ) is efficient iff Ai := Ai (0)⃗ ≡ {x; 0 < ̄ ̄ θi (x) = θ(x)} where θ(x) := max1≤j≤N θj (x). We observe that in the case of no commission, the efficiency condition is met if all agents set their flat prices to zero. In that case, the sum of all i. v.’s is maximized, and ⃗ ⃗ |m|⃗ ≤ μ(X)} = ∫[max θi (x)] dμ ≡ Ξθ,+ (0). Σθ := max{Σθ (m); X

i∈ℐ

+

Evidently, such an efficiency is not in the best interest of the agents (even though it is, of course, in the best interest of the consumers). An alternative definition, which is more realistic from the agent’s point of view, is the weak efficiency: The case of weak efficiency is obtained if all agents make a cartel, i. e., whenever all agents agree on a common price p̄ = pi for any i ∈ {1, . . . , N}. In that case the set of inactive consumers which does not attend any agent is A0 (p)̄ = {x; θi (x) − p̄ ≤ 0}. Definition 12.4.2. A subpartition A⃗ := (A1 , . . . , AN ) is weakly efficient iff there exists a common flat price p̄ such that any active consumer attends the agent best for him-

172 | 12 Sharing the individual value self, i. e., ̄ Ai := Ai (p,̄ . . . , p)̄ ≡ {x; p̄ < θi (x) = θ(x)}. It leaves the agents the freedom to choose the common price p.̄ If they choose p̄ in order to maximize the sum of their profits, then this p̄ is determined by the optimal price for a single agent whose utility function is θ:̄ p̄ = arg max pμ(x; θ(x) − p ≥ 0); p

see Example 12.1.1. If, on the other hand, the agents chose their common flat price p̄ in order to maximize the sum of their i. v., then, evidently, p̄ = min[θ(x)]+ , X

which leads to a strong efficiency. An additional, dual way to characterize a weakly efficient (sub)partitions is to ⃗ characterize a given total capacity m = |m|. ̄ Theorem 12.1. For any m < μ(X) there exists a weakly efficient subpartition A⃗ = (Ā 1 , . . . , Ā N ), μ(Ā i ) := m̄ i verifying ∑i m̄ i = m. The capacity vector m⃗ maximizes Σθ = ∑i Vi (mi ) on {m;⃗ ∑i mi ≤ m}, and the common price p̄ for this subpartition minimizes ̄ − p] dμ + pm. p 󳨃→ ∫[θ(x) + X

̄ Proof. Recall θ(x) := maxi θ(x) and ⃗ = min Ξθ,+ (p)⃗ + p⃗ ⋅ m.⃗ Σθ (m) ⃗ N p∈ℝ

̄ − p] ≥ [θ ] (x) − p for any i and any p ∈ ℝ, it follows from the definition of Since [θ(x) + i + θ,+ Ξ that for any m⃗ satisfying ∑Ni mi ≤ m, ̄ − p] dμ + pM ≥ Ξθ,+ (p1)⃗ + p1⃗ ⋅ m.⃗ ∫[θ(x) +

X

In particular, ̄ − p] dμ + pm ≥ min Ξθ,+ (p)⃗ + p⃗ ⋅ m⃗ = Σθ (m). ⃗ min ∫[θ(x) + p∈ℝ

⃗ N p∈ℝ

X

(12.6)

̄ ̄ Let Ā i = {x ∈ On the other hand, for the minimizer p̄ we get m = μ(x ∈ X; θ(x) > p). ̄ X; θi (x) > p}̄ and m̄ i := μ(Ai ). Then ∑i m̄ i = m and ̄ − p]̄ dμ + pm ̄ = ∑ ∫ θi dμ = Σθ (m̄ 1 , . . . , m̄ N ). ∫[θ(x) +

X

i

Ā i

This implies equality in (12.6) for m⃗ = (m̄ 1 , . . . , m̄ N ).

12.4 Efficiency | 173

To summarize: A weakly efficient (sub)partition is obtained either by a cartel sharing a common flat price or by maximizing the sum of the individual values subjected to a maximal total capacity ∑i mi ≤ m.

A natural question is the following: Is a weakly efficient (sub)partition guaranteed by maximizing the sum of agents’ profits (rather than the sum of their i. v.’s)?

Unfortunately, the answer to this question is negative, in general. Example 12.4.1. Consider the case where the supports of all agents’ utility functions are disjoint. The best price of agent i is then pi = arg max pμ{x; θi (x) ≥ p}. Evidently, there is no reason for all pi to be the same in that case! 12.4.1 Efficiency for agents of comparable utilities The opposite situation for Example 12.4.1 is whenever the support of all agents’ utilities are the same. A particular case is demonstrated in Example 11.1.3, where θi = λi θ, 0 ≤ λi < λi+1 . By Example 11.1.3, i 𝜕Σθ = ∑ λj (ℱθ󸀠 (ℳj ) − ℱθ󸀠 (ℳj+1 )) + λi ℱθ󸀠 (ℳi+1 ), 𝜕mi j=1

(12.7)

where we used ℳj := ∑Ni=j mi . By (12.3) we obtain that the sum of the profit of all agents, as a function of m,⃗ is ⃗ := ∑ 𝒫i (m) ⃗ = ∑ mi 𝒫 (m) i∈ℐ

i∈ℐ

i 𝜕Σθ = ∑ ∑ mi λj (ℱθ󸀠 (ℳj ) − ℱθ󸀠 (ℳj+1 )) 𝜕mi i∈ℐ j=1 N

N

+ ∑ mi λi ℱθ󸀠 (ℳi+1 ) = ∑ (∑ mi ) λj (ℱθ󸀠 (ℳj ) − ℱθ󸀠 (ℳj+1 )) i∈ℐ

j=1

i=j

N

+ ∑ mi λi ℱθ󸀠 (ℳi+1 ) = ∑ ℳj λj (ℱθ󸀠 (ℳj ) − ℱθ󸀠 (ℳj+1 )) i∈ℐ

j=1

+ ∑ (ℳi − ℳi+1 ) λi ℱθ󸀠 (ℳi+1 ) = ∑ ℳi (λi − λi−1 )ℱθ󸀠 (ℳi ), i∈ℐ

i∈ℐ

(12.8)

⃗ then follows for ℳi = ℳ0 for where we used λ0 = ℳN+1 ≡ 0. The maximum of 𝒫 (m) any i ∈ ℐ , where ℳ0 is the maximizer of m 󳨃→ mℱθ󸀠 (m). It implies that mN = ℳ0 and mi = 0 for 1 ≤ i < N. Thus we have the following:

174 | 12 Sharing the individual value Under the assumption of Example 11.1.3, the agents maximize the sum of their profits in the weakly effective state where all active consumers attend the leading agent N.

The cartel state in the last example is not necessarily a Nash equilibrium. Indeed, if p0 := ℱθ󸀠 (ℳ0 ) is the flat price of the leading agent N which maximizes his profit (as a single agent), then the cartel state is a Nash equilibrium iff the “second best” agent N − 1 cannot attract some consumers if she set her price higher than p0 , i. e., iff λN−1 max θ < p0 . X

(12.9)

Indeed, if this inequality is reversed, then agent N − 1 can set a price p0 < p < λN−1 maxX θ, attract the non-empty set of consumers AN−1 = {x; λN−1 θ(x) > p}, and gain a positive profit pμ(AN−1 ). We obtained the following theorem. Theorem 12.2. Under the conditions of Example 11.1.3, let p0 = ℱθ󸀠 (ℳ0 ), where ℳ0 is the maximizer of m 󳨃→ mℱθ (m) (equivalently, p0 is the minimizer of p 󳨃→ pμ{x; θ(x) > p}). Then the price vector pN = p0 , pi ≥ 0 for i < N is a Nash equilibrium under flat prices strategy iff (12.9) is satisfied. 12.4.2 Efficiency under commission strategy In general, however, it seems that under flat prices policy we cannot expect the cartel strategy leading to a maximal sum of the profit of the agents to be a (weakly) efficient state. The situation changes dramatically if the strategy of the agents involves commissions. Then efficiency can always be obtained if all agents make a cartel of zero flat prices p⃗ = 0 and a common commission qi = Q ∈ [0, 1). Indeed, in that case the (sub)partition is given by Ai = {x; 0 < (1 − Q)θi (x) = max (1 − Q)θj (x)}, 1≤j≤N

which is identical to Definition 12.4.1. It seems that the strategy of a cartel of commissions is a winning strategy for the agents. Fortunately (for the consumers), it is never a Nash equilibrium. Indeed, if all agents choose to collect a commission Q ≈ 1, then any agent can lower his commission a little bit and attract all consumers! What will be a Nash equilibrium in the case of Example 11.1.3 under a commission policy? Suppose the leading agent N sets up the commission qN = 1−λN−1 /λN . Then, for any choice qi ∈ (0, 1) for the other agents i ≠ N, the leading agent gets all consumers and her profit is (λN − λN−1 )μ(θ), while the profit of all other agents is zero. If agent N increases her commission even just a little bit, the next agent N − 1 may charge a sufficiently small (but positive) commission and win all the consumers! Since in the case qN = 1 − λN−1 /λN all agents except N get a zero profit anyway, they can set their commission arbitrarily at (0, 1).

12.5 Free price strategy | 175

In the case of Example 11.1.3, the Nash equilibrium for the “only commission” strategy is qN = 1 − λN−1 /λN and qi ∈ (0, 1) for i = 1, . . . , N − 1.

It seems, however, that an equilibrium in this class is not so safe for the leading agent N. Indeed, agent N − 1 may declare her commission qN−1 = 0. Even though she gains nothing from this choice, she competes with the leading agent N, since each consumer is now indifferent to the choice between N − 1 or N.2 Agent N − 1 may, then, try to negotiate with N for an agreement to share her profit. See Chapter 13.

12.5 Free price strategy Let us consider now the strategy by which each agent may choose her price arbitrarily: she is allowed to differentiate the consumers according to their utility functions with respect to all other agents. Let wi (x) be the charge of agent i from consumer x. The partition is now defined by A0 (w)⃗ := {x ∈ X; θi (x) − wi (x) ≤ 0, 1 ≤ i ≤ N},

⃗ Ai (w)⃗ := {x ∈ X; θi (x) − wi (x) > θj (x) − wj (x) ∀j ≠ i} − A0 (w). Note that if θi , wi are continuous functions, then Ai (w)⃗ are open sets for any i. The notion of Nash equilibrium is naturally generalized to the case of free strategies. However, the functions w⃗ 󳨃→ 𝒫i (w)⃗ := ∫ wi dμ ⃗ Ai (w)

are not continuous with respect to w⃗ ∈ C(X; ℝN+ ). Indeed, the dichotomy set {x; θj (x) − wj (x) = θi (x) − wi (x)}, i ≠ j, is not necessarily of measure zero for any admissible strategy w.⃗ This leads us to the following generalization. Definition 12.5.1. Let 𝒫i , i = 1, . . . , N, be defined and continuous on an open subset Q ⊂ C(X; ℝN+ ). Then w⃗ 0 ∈ Q is a weak Nash equilibrium if, for any sequence w⃗ n ∈ Q converging uniformly to w⃗ 0 , there exists a sequence of positive reals ϵn ↓ 0 such that 𝒫i (w⃗ n,−i , ζ ) ≤ 𝒫 (w⃗ n ) + ϵn

for any ζ ∈ C(X, ℝ+ ), i ∈ ℐ such that (w⃗ n,−i , ζ ) ∈ Q is the price strategy where agent i charges ζ (x) from a consumer x, while all other agents j ≠ i retain their prices wj . Such w⃗ 0 is efficient if, along such a sequence, μ(Ai (w⃗ n )ΔAi ) → 0 for i ∈ ℐ , where Ai is as given in Definition 12.4.1.3 2 Note that in that case, however, Assumption 12.0.1 is not met. 3 Here AΔB := (A − B) ∪ (B − A) is the symmetric difference.

176 | 12 Sharing the individual value Another formulation of the weak Nash equilibrium is presented in the following box. w⃗ 0 is a weak Nash equilibrium iff for any ϵ > 0 there exists an ϵ-neighborhood of w⃗ 0 such that for any admissible strategy w⃗ in this neighborhood, no agent can improve her reward more than ϵ by changing the price she collects, provided all other agents retain their pricing w.⃗

The free price strategy contains, as special cases, the flat price strategy wi (x) = pi , the commission strategy wi (x) = qi θi (x), qi ∈ (0, 1], and the mixed strategy wi (x) = pi + qi θi (x). We recall that the existence of a (pure strategy) Nash equilibrium is not guaranteed, in the general case, for either the flat price, commission or mixed strategies. Moreover, even in the case where such a Nash equilibrium exists, it is not efficient, in general. In the case of a free price strategy, however, we can guarantee the existence of a weak Nash equilibrium which is efficient.

12.5.1 Where Nash equilibrium meets efficiency Let wi (x) := {

θi (x) − maxj=i̸ θj (x) 0

if x ∈ Ai , if x ∈ ̸ Ai ,

(12.10)

where Ai is as given in Definition 12.4.1. Under the strategy (12.10), any consumer x obtains the utility of his “best next agent,” that is, max θj (x),

j =i(x) ̸

where i(x) := arg max1≤j≤N θj (x).

We leave the reader to prove the following theorem. Theorem 12.3. If μ{x; θi (x) = θj (x) = 0} = 0 for any i ≠ j, then the free strategy (12.10) is an efficient Nash equilibrium (in the sense of Definition 12.5.1). The free strategy seems to be good news for the consumers. At least, it guarantees that each consumer will get the utility of his next best agents, and verifies both the stability under competitive behavior (in the sense that the weak Nash equilibrium condition is satisfied) and efficiency. In the next chapter we shall see, however, that this strategy does not survive a cooperative behavior of the agents.

13 Cooperative partitions Competition has been shown to be useful up to a certain point and no further, but cooperation, which is the thing we must strive for today, begins where competition leaves off. (F.D.R.)

13.1 Free price strategy Using a free price strategy discussed in Section 12.5, we obtained a weak Nash equilibrium which is efficient by Theorem 12.3. However, the agents may beat this strategy by forming a coalition. Let us elaborate this point. Suppose that some agents 𝒥 ⊂ ℐ := {1, . . . , N} decide to establish a coalition: they offer any client x the maximal utility of the coalition members θ𝒥 (x) := max θi (x).

(13.1)

i∈𝒥

So, the “superagent” 𝒥 is now competing against the other agents ℐ − 𝒥 . The efficient partition of X now takes the form A𝒥 := {x ∈ X; θ𝒥 (x) > [ max θi (x)] } = ⋃ Ai , i∈ℐ−𝒥

+

i∈𝒥

(13.2)

where Ai is as given in Definition 12.4.1. The 𝒥 -component of the free price strategy (12.10) corresponding to the set of agents {θ𝒥 , θi , i ∈ ̸ 𝒥 } is, by Theorem 12.3, w𝒥 (x) := {

θ𝒥 (x) − maxj=𝒥 ̸ θj (x) any positive value

if x ∈ A𝒥 , if x ∈ ̸ A𝒥 .

(13.3)

Clearly, w𝒥 (x) ≥ wj (x) for any x ∈ A𝒥 and any j ∈ 𝒥 . In particular, the profit of the superagent 𝒥 (denoted as ν(𝒥 )) is not smaller than the combined profits of all agents j ∈ 𝒥 together (under the free price strategy): ν(𝒥 ) := ∫ w𝒥 dμ ≥ ∑ ∫ wj dμ. A𝒥

j∈𝒥 A

(13.4)

j

The inequality in (13.4) can be strong. Evidently, this profit is monotone in the coalition, namely, ν(𝒥 󸀠 ) ≥ ν(𝒥 ) whenever 𝒥 󸀠 ⊃ 𝒥 . In particular, if 𝒥 = ℐ (the grand coalition), then wℐ = θ̄+ ≡ maxi∈ℐ [θi ]+ . In that case the grand coalition of agents wins the whole surplus value ν(ℐ ) = ∫X θ̄+ dμ, and, in particular, we get an efficient partition. Is the grand coalition, indeed, a stable position for the agents? It depends on how the agents share the surplus value between themselves. A natural way of sharing is as https://doi.org/10.1515/9783110635485-013

178 | 13 Cooperative partitions follows: each agent collects the surplus value in the domain in which she dominates, that is, 𝒫i = ∫ θ̄+ dμ ≡ ∫[θi ]+ dμ Ai

Ai

(recall Definition 12.4.1). Note that the agents may almost obtain such a sharing if they act individually, and use the commission strategy wi = qθi , q ∈ (0, 1) for sufficiently small 1 − q. However, such a sharing it is not a Nash equilibrium by the argument in Section 12.4.2, as any agent may slightly lower her commission and attract the consumers of other agents. At this point we leave the realm of Nash equilibrium and competitive game theory and enter into the realm of cooperative games.

13.2 Cooperative games – a crash review A cooperative game is a game where groups of players (“coalitions”) may enforce cooperative behavior, hence the game is a competition between coalitions of players, rather than between individual players. This section is based on the monograph [20]. Definition 13.2.1. A cooperative game in ℐ := {1, . . . , N} is given by a reward function ν on the subsets of ℐ : ν : 2ℐ → ℝ+ ,

ν(0) = 0.

The set of imputations is composed of vectors x⃗ := (x1 , . . . , xN ) ∈ ℝN+ which satisfy the following conditions: ∑ xi ≤ ν(ℐ ).

i∈ℐ

(13.5)

Definition 13.2.2. The core of a game ν : 2ℐ → ℝ+ (Core(ν)) is the collection of all imputation vectors which satisfy ∀𝒥 ⊆ ℐ ,

∑ xj ≥ ν(𝒥 ).

i∈𝒥

(13.6)

If the core is not empty, then no subcoalition 𝒥 of the grand coalition ℐ will be formed. Indeed, if such a subcoalition 𝒥 is formed, its reward ν(𝒥 ) is not larger than the sum of the imputations of its members, guaranteed by the grand coalition. In many cases, however, the core is empty. We can easily find a necessary condition for the core to be non-empty. Suppose we divide ℐ into a set of coalitions 𝒥k ⊂ ℐ , k = 1, . . . , m, such that 𝒥k ∩ 𝒥k󸀠 = 0 for k ≠ k 󸀠 and ⋃m k=1 𝒥j = ℐ .

13.2 Cooperative games – a crash review

| 179

Proposition 13.1. For any such division, the condition m

∑ ν(𝒥k ) ≤ ν(ℐ )

k=1

(13.7)

is necessary for the grand coalition to be stable. Proof. Suppose x⃗ ∈ Core(ν). Let ν(̃ 𝒥 ) := ∑i∈𝒥 xi . Then ν(̃ 𝒥 ) ≥ ν(𝒥 ) for any 𝒥 ⊆ ℐ . If ̃ 𝒥k ) ≥ ∑m (13.7) is violated for some division {𝒥1 , . . . , 𝒥m }, then ∑m k=1 ν( k=1 ν(𝒥k ) > ν(ℐ ). m ̃ On the other hand, ∑k=1 ν(𝒥k ) = ∑i∈ℐ xi ≤ ν(ℐ ), so we get a contradiction. Note that superadditivity ν(𝒥1 ) + ν(𝒥2 ) ≤ ν(𝒥1 ∪ 𝒥2 )

∀ 𝒥1 ∩ 𝒥2 = 0

(13.8)

is a sufficient condition for (13.7). However, (13.8) by itself is not a sufficient condition for the stability of the grand coalition. Example 13.2.1. In the case N = 3 the game ν(1) = ν(2) = ν(3) = 0, ν(12) = ν(23) = ν(13) = 3/4, ν(123) = 1 is superadditive but its core is empty. We may extend condition (13.7) as follows: A weak division is a function λ : 2ℐ → ℝ which satisfies the following: i) For any 𝒥 ⊆ {1, . . . , N}, λ(𝒥 ) ≥ 0. ii) For any i ∈ ℐ , ∑𝒥 ⊆ℐ;i∈𝒥 λ(𝒥 ) = 1. A collection of such sets {𝒥 ⊂ ℐ ; λ(𝒥 ) > 0} verifying (i), (ii) is called a balanced collection [20]. We can think about λ(𝒥 ) as the probability of the coalition 𝒥 . In particular, (ii) asserts that any individual i ∈ ℐ has a probability 1 to belong to some coalition 𝒥 . Note that any division {𝒥1 , . . . , 𝒥m } is, in particular, a weak division where λ(𝒥 ) = 1 if 𝒥 ∈ {𝒥1 , . . . , 𝒥m } and λ(𝒥 ) = 0 otherwise. It is not difficult to extend the necessary condition (13.7) to weak subdivisions as follows. Proposition 13.2. For any weak subdivision λ, the condition ∑ λ(𝒥 )ν(𝒥 ) ≤ ν(ℐ )

(13.9)

𝒥 ∈2ℐ

is necessary for the grand coalition to be stable. The proof of Proposition 13.2 is a slight modification of the proof of Proposition 13.1. However, it turns out that (13.9) is also a sufficient condition for the stability of the grand coalition ℐ . This is the content of the Bondareva–Shapley theorem.

180 | 13 Cooperative partitions Theorem 13.1 ([8, 44]). The grand coalition is stable iff it satisfies (13.9) for any weak division λ. The condition of Theorem 13.1 is easily verified for superadditive games in the case N = 3. Corollary 13.2.1. A superadditive cooperative game of three agents (N = 3) admits a non-empty core iff ν(12) + ν(13) + ν(23) < 2ν(123).

(13.10)

Indeed, it can be shown that all weak subdivisions for N = 3 are spanned by λ(𝒥 ) = 1/2 if 𝒥 = (12), (13), (23);

λ(𝒥 ) = 0 otherwise,

and the trivial ones.

13.2.1 Convex games A game ν is said to be convex if a larger coalition gains from joining a new agent at least as much as a smaller coalition gains from adding the same agent. That is, if 𝒥2 ⊃ 𝒥1 and {i} ∈ ̸ 𝒥1 ∪ 𝒥2 , then ν(𝒥2 ∪ {i}) − ν(𝒥2 ) ≥ ν(𝒥1 ∪ {i}) − ν(𝒥1 ).

(13.11)

Inequality (13.11) follows if for any 𝒥1 , 𝒥2 ∈ 2ℐ ν(𝒥1 ) + ν(𝒥2 ) ≤ ν(𝒥1 ∪ 𝒥2 ) + ν(𝒥1 ∩ 𝒥2 ).

(13.12)

In fact, it turns out that (13.11) and (13.12) are equivalent. The last condition is called supermodular (Section 7.4 in [36]). Note that supermodularity is stronger than superadditivity (13.8). However, in contrast to superadditivity, supermodularity does imply the existence of a non-empty core. Moreover, it characterizes the core in a particular, neat way: Let i1 , . . . , iN be any arrangement of the set ℐ . For each such arrangement, consider the imputations: xi1 = ν({i1 }), . . . , xik = ν({i1 , . . . , ik }) − ν({i1 , . . . , ik−1 }) . . . .

(13.13)

Theorem 13.2 ([15]). If the game is convex, then any imputation (13.13) obtained from an arbitrary arrangement of the agents is in the core. Moreover, the core is the convex hull of all such imputations.

13.2 Cooperative games – a crash review

| 181

Example 13.2.2. Let (X,̄ μ)̄ be a measure space, μ(X)̄ < ∞. Let us associate with each agent i ∈ ℐ a measurable set Ā i ⊂ X.̄ For any 𝒥 ⊂ ℐ let ν(𝒥 ) := μ̄ (X̄ − ⋃ Ā j ) . j∈𝒥 ̸

Lemma 13.1. The game defined in Example 13.2.2 is convex. Proof. By the postulates of measure ν(𝒥 ) = μ(̄ X)̄ − μ(̄ Ā ℐ−𝒥 ), where Ā 𝒥 := ⋃j∈𝒥 Ā j . Then Ā ℐ−(𝒥1 ∪𝒥2 ) ⊂ Ā ℐ−𝒥1 ∩ Ā ℐ−𝒥2 . Indeed, x ∈ Ā ℐ−(𝒥1 ∪𝒥2 ) iff there exists i ∈ ℐ − (𝒥1 ∪ 𝒥2 ) such that x ∈ Ā i , which implies that x ∈ Ā ℐ−𝒥1 ∩ Ā ℐ−𝒥2 . This inclusion can be strict since x ∈ Ā ℐ−𝒥1 ∩ Ā ℐ−𝒥2 implies that there exists i ∈ ℐ − 𝒥1 and j ∈ ℐ − 𝒥2 such that x ∈ Ai ∩ Aj (but not necessarily i = j). On the other hand, Ā ℐ−(𝒥1 ∩𝒥2 ) = Ā ℐ−𝒥1 ∪ Ā ℐ−𝒥2 .

(13.14)

μ̄ (Ā ℐ−(𝒥1 ∪𝒥2 ) ) ≤ μ̄ (Ā ℐ−𝒥1 ∩ Ā ℐ−𝒥2 )

(13.15)

Hence

and μ̄ (Ā ℐ−(𝒥1 ∩𝒥2 ) ) = μ̄ (Ā ℐ−𝒥1 ∪ Ā ℐ−𝒥2 ) . By the axioms of a measure we also get μ̄ (Ā ℐ−𝒥1 ∪ Ā ℐ−𝒥2 ) = μ̄ (Ā ℐ−𝒥1 ) + μ̄ (Ā ℐ−𝒥2 ) − μ̄ (Ā ℐ−𝒥1 ∩ Ā ℐ−𝒥2 ) . Since μ̄ (Ā ℐ−𝒥1 ∪ Ā ℐ−𝒥2 ) ≡ μ̄ (Ā ℐ−(𝒥1 ∩𝒥2 ) ) ≡ μ(̄ X)̄ − ν(𝒥1 ∩ 𝒥2 ) and μ̄ (Ā ℐ−𝒥1 ∩ Ā ℐ−𝒥2 ) ≥ μ̄ (Ā ℐ−(𝒥1 ∪𝒥2 ) ) ≡ μ(̄ X)̄ − ν(𝒥1 ∪ 𝒥2 ), we obtain ν (𝒥1 ∪ 𝒥2 ) + ν (𝒥1 ∩ 𝒥2 ) ≥ ν (𝒥1 ) + ν (𝒥2 ) .

182 | 13 Cooperative partitions

13.3 Back to cooperative partition games Let us re-examine the game described in Section 13.1. Here we defined ν(𝒥 ) := ∫ w𝒥 dμ

(13.16)

A𝒥

(see (13.4)), where A𝒥 , w𝒥 are as in (13.2), (13.3). Let us extend the space X to the graph below the maximal utility function θ̄+ , that is, X̄ := {(x, s); x ∈ X, 0 ≤ s ≤ θ̄+ (x) := max[θi (x)]+ }. i∈ℐ

Let us further define Ā j := {(x, s) ∈ X;̄ 0 ≤ s ≤ [θj (x)]+ }. It follows that the game (13.16) is equivalent, under this setting, to the game described ̄ in Example 13.2.2 where μ(dxds) = μ(dx)ds. From Lemma 13.1 and Theorem 13.2 we obtain the following: Theorem 13.3. Under condition of Theorem 12.3, the cooperative game of free price (13.16) is stable. This is good news for the agents but very bad for the consumers! Indeed, the stable grand coalition of the agents collects all the surplus to themselves (as ν(ℐ ) = ∫X θ̄+ dx) and leaves nothing to the consumers. In order to defend the consumers we have to impose some regulation on the agents: Consumer-based pricing is forbidden!

13.3.1 Flat prices strategy: regulation by capacity Let us assume now that each agent has a limited capacity. So, μ(Ai ) ≤ m0i , where Ai ⊂ X is the set of consumers of agent i. The agents may still form a coalition 𝒥 ⊂ ℐ , and the capacity of 𝒥 is just m0𝒥 := ∑ m0i . i∈𝒥

The utility of the coalition 𝒥 is given by maximizing the utilities of its members, i. e., θ𝒥 is as defined (13.1).

13.3 Back to cooperative partition games | 183

We assume that for any coalition 𝒥 ⊂ ℐ, the rest of the agents form the complement coalition 𝒥 − := ℐ − 𝒥 .

Let us consider a cooperative game ν where the utility of a coalition ν(𝒥 ) is the surplus value of this coalition, competing against the complement coalition 𝒥 − . For this we consider Ξ𝒥 (p𝒥 , p𝒥 − ) := ∫ max [(θ𝒥 (x) − p𝒥 ), θ𝒥 − (x) − p𝒥 − ), 0] dμ(x) : ℝ2 → ℝ,

(13.17)

X

and Σ(m0𝒥 , m0𝒥 − ) =

max

m𝒥 ≤m0𝒥 ,m𝒥 − ≤m0

𝒥−

[

min

(p𝒥 ,p𝒥 − )∈ℝ2

Ξ𝒥 (p𝒥 , p𝒥 − ) + p𝒥 m𝒥 + p𝒥 − m𝒥 − ] . (13.18)

Proposition 13.3. Under the assumption of Theorem 11.1, there exist unique vectors (m𝒥 , m𝒥 − ) which maximize (13.18) and unique (p0𝒥 , p0𝒥 − ) which minimize (p𝒥 , p𝒥 − ) 󳨃→ Ξ𝒥 (pJ , p𝒥 − ) + pJ m𝒥 + pJ − m𝒥 − . Proof. First note that {x ∈ X; θJ (x) − θJ − (x) = r} ⊂ ⋃ {x ∈ X; θj (x) − θi (x) = r}, j∈J,i∈J −

so Assumption 11.0.1 implies, for any r ∈ ℝ, μ(x; θ𝒥 (x) − θ𝒥 − (x) = 0) = 0;

μ(x; θ𝒥 (x) = 0) = μ(x; θ𝒥 − (x) = 0) = 0.

Hence, the conditions of Theorem 11.1 hold for this modified setting. The partition (A0𝒥 , A0𝒥 − ) is also given by A0𝒥 := ⋃ Ai (p⃗ 0 ), i∈𝒥

where Ai (p)⃗ as defined in (11.5) and p0,i = p0𝒥 if i ∈ 𝒥 and p0,j = p0𝒥 − if j ∈ ̸ 𝒥 . Indeed, Ξθ,+ (p⃗ 0 ) ≡ Ξ𝒥 (p0𝒥 , p0𝒥 − ), where Ξθ,+ is given by (11.2). Thus, we may characterize the coalitions 𝒥 as a cartel: The coalition 𝒥 is obtained as a cartel where all members of this coalition (and, simultaneously, all members of the complementary coalition 𝒥 − ) agree on equal flat prices.

Definition 13.3.1. Let A0𝒥 := A𝒥 (p0J , p0𝒥 − ), where AJ (p𝒥 , p𝒥 − ) := {x ∈ X; θ𝒥 (x) − p𝒥 ≥ (θ𝒥 − (x) − p𝒥 − )+ } and (p0J , p0𝒥 − ) is the unique minimizer as defined in Proposition 13.3.

184 | 13 Cooperative partitions The surplus-based coalition game ν subjected to a given capacity vector m⃗ 0 ∈ ℝN+ is given by ν(𝒥 ) := ∫ θ𝒥 dμ. A0𝒥

Note that this game satisfies the following condition: For each J ⊂ ℐ , ν(𝒥 ) + ν(𝒥 − ) ≤ ν(ℐ )

∀ 𝒥 ⊂ ℐ,

(13.19)

which is a necessary condition for superadditivity (13.8). In general, however, thus game is not supermodular. Example 13.3.1. Let us consider three agents corresponding to θ1 ≥ θ2 ≥ θ3 . Assume also m1 , m2

m1 + m2 θ1 (x0 ), m1

then ν({1, 2}) < ν({1}) ≤ ν({1}) + ν({2}). An alternative definition of a coalition game is based on the agent’s profit. In that case there is an upper limit to the capacity of all agents, and each coalition 𝒥 maximizes its profit against the complement coalition 𝒥 − . Definition 13.3.2. Let m > 0. Given a coalition 𝒥 ⊂ ℐ , define the self-profit coalition game as ν𝒫 (𝒥 ) := m𝒥

𝜕 Σ(m𝒥 , m𝒥 − ), 𝜕m𝒥

where Σ is as defined in (13.18). Recall that ν𝒫 (𝒥 )/m𝒥 stands for the flat price of the first (super)agent 𝒥 . Surely we cannot expect the self-profit game to be superadditive, in general. In fact, inequality (13.19) is not necessarily valid for such a game, even in the case of only two agents (Example 12.4.1).

13.3 Back to cooperative partition games | 185

13.3.2 Coalition games under comparable utilities We obtained that both coalition games given by Definitions 13.3.1 and 13.3.2 are not superadditive in general. However, there is a special case, introduced in Example 11.1.3, for which we can guarantee superadditivity and, moreover, even stability under certain additional conditions (Example 11.1.3). Assumption 13.3.1. There exists a non-negative θ : X ∈ C(X) satisfying μ(x; θ(x) = r) = 0 for any r ∈ ℝ. The utilities θi are given by θi = λi θ, where λ⃗ := (λ1 , . . . , λN ) ∈ ℝN+ such that 0 < λ1 < ⋅ ⋅ ⋅ < λN . Proposition 13.4. Under Assumption 13.3.1, for any m⃗ := (m1 , . . . , mN ) ∈ ℝN+ , the surplus-based game ν is superadditive. If, in addition, m 󳨃→ m(ℱθ (m))󸀠 is monotone non-decreasing on [0, M] (Example 11.1.1), then the profit-based game is superadditive as well, provided ∑i∈ℐ mi ≤ M. Proof. Surplus-based game: From Example 11.1.3 (in particular from (11.27)) we obtain that the surplus value of agent i under optimal partition is Vi ≡ λi (ℱθ (ℳi ) − ℱθ (ℳi+1 )) .

(13.20)

ν(𝒥 ) = λN ℱθ (m𝒥 ),

(13.21)

ν(𝒥 ) = λ𝒥 (ℱθ (M) − ℱθ (M − m𝒥 )) ,

(13.22)

It follows that if {N} ∈ 𝒥 ,

while if {N} ∈ ̸ 𝒥 ,

where λ𝒥 = maxi∈𝒥 λi < λN and M = m𝒥 + m𝒥 − ≡ ∑i∈ℐ mi . Let now 𝒥1 , 𝒥2 ⊂ ℐ such that 𝒥1 ∩ 𝒥2 = 0 (in particular, m𝒥1 + m𝒥2 ≤ M). Assume first {N} ∈ ̸ 𝒥1 ∪ 𝒥2 . Then from (13.22) ν(𝒥1 ∪ 𝒥2 ) = λ𝒥1 ∨ λ𝒥2 (ℱθ (M) − ℱθ (M − m𝒥1 ∪𝒥2 ))

= λ𝒥1 ∨ λ𝒥2 (ℱθ (M ∗ ) − ℱθ (M − m𝒥1 − m𝒥2 )) .

Now, ℱθ (M) − ℱθ (M − m𝒥1 − m𝒥2 ) ≥ 2ℱθ (M) − ℱθ (M − m𝒥1 ) − ℱθ (M − m𝒥2 ),

since ℱθ (M) − ℱθ (M − m𝒥1 ) ≤ ℱθ (M − m𝒥2 ) − ℱθ (M − m𝒥1 − m𝒥2 )

(13.23)

186 | 13 Cooperative partitions by the concavity of ℱθ . It follows from (13.23) that ν(𝒥1 ∪ 𝒥2 ) ≥ λ𝒥1 ∨ λ𝒥2 [(ℱθ (M) − ℱθ (M − m𝒥1 )) + (ℱθ (M) − ℱθ (M − m𝒥2 ))]

≥ λ𝒥1 (ℱθ (M) − ℱθ (M − m𝒥1 )) + λ𝒥2 (ℱθ (M) − ℱθ (M − m𝒥2 )) = ν(𝒥1 ) + ν(𝒥2 ).

Next, if, say, {N} ∈ 𝒥1 , then, using (13.21), (13.22), ν(𝒥1 ∪ 𝒥2 ) = λN ℱθ (m𝒥1 + m𝒥2 ), ν(𝒥1 ) = λN ℱθ (m𝒥1 ), ν(𝒥2 ) = λ𝒥2 (ℱθ (M) − ℱθ (M − m𝒥2 )) , so ν(𝒥1 ∪ 𝒥2 ) − ν(𝒥1 ) − ν(𝒥2 ) ≥ λN [ℱθ (m𝒥1 + m𝒥2 ) − ℱθ (m𝒥1 ) − ℱθ (M) + ℱθ (M − m𝒥2 )] ≥ 0, again, by concavity of ℱθ and since M ≥ m𝒥1 + m𝒥2 . Profit-based game: From (11.27) with the two agents (λ1 θ, m1 ), (λ2 θ, m2 ) where λ2 > λ2 we get Σθ (m1 , m2 ) = λ1 (ℱθ (m1 + m2 ) − ℱθ (m2 )) + λ2 ℱθ (m2 ). Assume first N ∈ ̸ 𝒥1 ∪𝒥2 . Then, we substitute (m1 , m2 ) for (m𝒥1 , M−m𝒥1 ), (m𝒥2 , M−m𝒥2 ), and (m𝒥1 ∪𝒥2 , 1 − m(𝒥1 ∪𝒥2 ) ) and we get ν𝒫 (𝒥1 ) = m𝒥1 λ𝒥1 (ℱθ )󸀠 (M), ν𝒫 (𝒥2 ) = m𝒥2 λ𝒥2 (ℱθ )󸀠 (M) and ν𝒫 (𝒥1 ∪ 𝒥2 ) = m𝒥1 ∪𝒥2 λ𝒥1 ∪𝒥2 (ℱθ )󸀠 (M) ≡ λ𝒥1 ∨ λ𝒥2 (m𝒥1 + m𝒥2 )(ℱθ )󸀠 (M). In particular, we obtain ν𝒫 (𝒥1 ∪𝒥2 )−ν𝒫 (𝒥1 )−ν𝒫 (𝒥2 ) = (λ𝒥1 ∨ λ𝒥2 (m𝒥1 + m𝒥2 ) − λ𝒥1 m𝒥1 − λ𝒥2 m𝒥2 ) (ℱθ )󸀠 (M) > 0 (unconditionally!). Assume now that N ∈ 𝒥2 . In particular, λN > λ𝒥1 . Thus, under the same setting: ν𝒫 (𝒥1 ) = m1 𝜕Σθ /𝜕m1 = m𝒥1 λ𝒥1 (ℱθ )󸀠 (M), ν𝒫 (𝒥2 ) = m2 𝜕Σθ /𝜕m2 = m𝒥2 (λ𝒥1 (ℱθ )󸀠 (M) + (λN − λ𝒥1 ) (ℱθ )󸀠 (m𝒥2 )) , ν𝒫 (𝒥1 ∪ 𝒥2 ) = m𝒥1 ∪𝒥2 (λ𝒥1 (ℱθ )󸀠 (M) + (λ𝒥2 − λ𝒥1 ) (ℱθ )󸀠 (m𝒥1 ∪𝒥2 ))

= (m𝒥1 + m𝒥2 ) (λ𝒥1 (ℱθ )󸀠 (M) + (λN − λ𝒥1 ) (ℱθ )󸀠 (m𝒥1 + m𝒥2 )) .

It follows that ν𝒫 (𝒥1 ∪ 𝒥2 ) − ν𝒫 (𝒥1 ) − ν𝒫 (𝒥2 ) = (λ𝒥2 − λ𝒥1 ) ((m𝒥1 + m𝒥2 ) (ℱθ )󸀠 (m𝒥1 + m𝒥2 ) − m𝒥2 (ℱθ )󸀠 (m𝒥2 )) ≥ 0 by the assumption of monotonicity of m 󳨃→ m(Fθ∗ )󸀠 (m) on [0, M], and m𝒥 , m𝒥 − ∈ [0, M].

13.3 Back to cooperative partition games | 187

Under the assumption of Proposition 13.4 we may guess, intuitively, that the grand coalition is stable if the gap between the utilities of the agents is sufficiently large (so the other agents are motivated to join the smartest one), and the capacity of the wisest agent (N) is sufficiently small (so she is motivated to join the others as well). Below we prove this intuition in the case N = 3. Proposition 13.5. Under the assumption of Proposition 13.4 and N = 3, λ3 ℱθ (m1 + m2 ) > λ2 ℱθ (m2 ) + ℱθ (m1 ) is necessary and sufficient for the stability of the grand coalition in the surplus game. Here ℱθ is as defined in Example 11.1.3. Proof. From Corollary 13.2.1 and Proposition 13.4 we only have to prove (13.10). Now, ν(123) = λ3 ℱθ (μ(X)), ν(13) = λ3 (ℱθ (μ(X)) − ℱθ (m2 )), ν(23) = λ3 (ℱθ (μ(X)) − ℱθ (m1 )), and ν(12) = λ2 ℱθ (m1 + m2 ). The result follows from substituting the above in (13.10). Theorem 13.4. Assume m 󳨃→ m(ℱθ )󸀠 (m) is non-decreasing on [0, M], where M = m1 + m2 + m3 . Assume further that α (ℱθ )󸀠 (m2 ) + β (ℱθ )󸀠 (m1 ) < (ℱθ )󸀠 (m1 + m2 + m3 ), where α := 2m (λ 1

β :=

(m1 +m3 )(λ3 −λ2 )

3 −λ2 )+(m2 +m3 )(2λ3 −λ2 )

(13.24)

,

(m2 +m3 )(λ3 −λ1 ) . 2m1 (λ3 −λ2 )+(m2 +m3 )(2λ3 −λ2 )

Then the self-profit game ν as given in Definition 13.3.2 is stable. Recall that ℱθ is a concave function, and hence (ℱθ )󸀠 (m1 + m2 + m3 ) is smaller than both (ℱθ )󸀠 (m1 ) and (ℱθ )󸀠 (m2 ). Hence 0 < α + β < 1 is a necessary condition for (13.24). Check that this condition is always satisfied (since λ3 > λ2 ). Proof. Again, the superadditivity is given by Proposition 13.4. We have ν𝒫 (123) = (m1 + m2 + m3 )λ3 (ℱθ )󸀠 (m1 + m2 + m3 ),

ν(13) = (m1 + m3 ) [λ2 (ℱθ )󸀠 (m1 + m2 + m3 ) + (λ3 − λ2 ) (ℱθ )󸀠 (m2 )] , ν(23) = (m2 + m3 ) [λ1 (ℱθ )󸀠 (m1 + m2 + m3 ) + (λ3 − λ1 ) (ℱθ )󸀠 (m1 )] ,

while ν𝒫 (12) = (m1 + m2 )λ2 (ℱθ )󸀠 (m1 + m2 + m3 ). Thus 2ν𝒫 (123) − ν(12) − ν(13) − ν(23)

188 | 13 Cooperative partitions = (ℱθ )󸀠 (m1 + m2 + m3 ) [2m1 (λ3 − λ2 ) + (m2 + m3 )(2λ3 − λ2 )]

− (m1 + m3 )(λ3 − λ2 ) (ℱθ )󸀠 (m2 ) − (m2 + m3 )(λ3 − λ1 ) (ℱθ )󸀠 (m1 )

and the result follows by (13.10) as well.

A Convexity For the completeness of exposition we introduce some basic notions from the theory of convexity. We only consider linear spaces 𝕄󸀠 over the reals ℝ of finite dimension. This restriction, which is sufficient for our purpose, will render the reference to any topology. In fact, topology enters only through the definition of the dual space of 𝕄󸀠 , 𝕄+ , that is, the space of all continuous linear functionals on 𝕄󸀠 , and we denote the duality pairing by ⃗ : 𝕄󸀠 × 𝕄+ → ℝ. (P⃗ : M) Since, as is well known, all norms are equivalent on a linear space of finite dimension, it follows that the notion of a continuous functional is norm-independent. Even though we distinguish between the space 𝕄󸀠 and its dual 𝕄+ (which are isomorphic), we do not distinguish weak, weak*, and strong (norm) convergence of sequences in the spaces 𝕄󸀠 and 𝕄+ , respectively. The notion of open, closed sets and interior, cluster points of sets are defined naturally in terms of a generic norm.

A.1 Convex sets The notion of a convex set is pretty natural: A set C ⊂ 𝕄󸀠 is convex iff for any P⃗ 1 , P⃗ 2 ∈ A, the interval connection P⃗ 1 , P⃗ 2 is contained in C. That is, sP⃗ 1 + (1 − s)P⃗ 2 ∈ C for any s ∈ [0, 1]. Note that a convex set may be open, closed, or neither. A convex set is called strictly convex if for any two points P⃗ 1 , P⃗ 2 ∈ C, the open interval sP⃗ 1 + (1 − s)P⃗ 2 , s ∈ (0, 1) is contained in the interior of C. In particular, a convex set whose interior is empty is not strictly convex. For example, if C is contained in a subspace of L ⊂ 𝕄+ are not strictly convex. More generally, if the boundary of a convex set contains an open set in the relative topology of a subspace, then it is not strictly convex. P⃗ ∈ C is an extreme point iff it is not contained in the interior of any interval contained in C, i. e., there exists no P⃗ 1 ≠ P⃗ 2 both in C and α ∈ (0, 1) such that P⃗ = αP⃗ 1 + (1 − α)P⃗ 2 . Examples of extreme points are the boundary of a solid ellipsoid, or the vertices of a Polyhedron. A stronger notion is that of exposed points. A point is an exposed point of C if there exists a linear functional such that this point is the unique maximizer of the functional on C. Alternatively, there exists a codimensional 1 hyperplane whose intersection with C is this single point. See Fig A.1. Proposition A.1. 1. The closure and the interior of a convex set are convex. 2. The intersection of any number of convex sets is convex. https://doi.org/10.1515/9783110635485-014

190 | A Convexity

Figure A.1: Left: Strictly convex set. All boundary points are exposed. Right: A (not strictly) convex set. Exposed points are marked by bold dots.

3.

If the interior of a convex set C is not empty, then the closure of the interior of C is the closure of C.

Definition A.1.1. The convex hull of a set A (Conv(A)) is the intersection of all convex sets containing A. In particular, it is the minimal convex set containing A. An equivalent definition of the convex hull is obtained in terms of the convex combinations: A convex combination of points x1 , . . . , xk , k ∈ ℕ, is a point x = ∑ki=1 λi xi , where λi ≥ 0 and ∑ki=1 λi = 1. Lemma A.1. The convex hull of a set A is the set of all convex combinations of its points. A fundamental theorem is the Krein–Milman theorem. Theorem A.1 ([30]). Any convex set is the convex hull of its extreme points. The Krein–Milman theorem is valid in much wider cases, namely, for any set in a Hausdorff locally convex topological vector space. In particular, it is valid for the set of Borel measures in compact metric space.

A.2 Convex functions The basic notion we consider is that of a convex function Ξ : 𝕄󸀠 → ℝ ∪ {∞} := ℝ.̂ The fundamental definition is the following. Definition A.2.1. Ξ is a convex function on 𝕄󸀠 if for any P⃗ 1 , P⃗ 2 ∈ 𝕄󸀠 and any s ∈ [0, 1]: Ξ(sP⃗ 1 + (1 − s)P⃗ 2 ) ≤ sΞ(P⃗ 1 ) + (1 − s)Ξ(P⃗ 2 ). Ξ is strictly convex at P⃗ 0 if for any P⃗ 1 ≠ P⃗ 2 and s ∈ (0, 1) such that P⃗ 0 := sP⃗ 1 + (1 − s)P⃗ 2 Ξ(P⃗ 0 ) < sΞ(P⃗ 1 ) + (1 − s)Ξ(P⃗ 2 ).

A.2 Convex functions |

191

Note that we allow Ξ to obtain the value {∞} (but not the value {−∞}), and we use, of course, the rule r + ∞ = ∞ for any r ∈ ℝ. The essential domain of Ξ (ED(Ξ)) is the set on which Ξ admits finite values: ED(Ξ) := {P⃗ ∈ 𝕄󸀠 ; Ξ(P)⃗ ∈ ℝ}. Remark A.2.1. In this book we are usually assuming that Ξ is real-valued for any P⃗ ∈ 𝕄󸀠 (i. e., ED(Ξ) = 𝕄󸀠 ). This, however, is not true for the Legendre transform of Ξ defined below on the dual space 𝕄+ . Since we treat (𝕄󸀠 , Ξ) and (𝕄+ , Ξ∗ ) on the same footing, we allow Ξ to take infinite values as well. There are two natural connections between convex functions and convex sets, as well as between points of strict convexity and extreme points. The first corresponds to the definition of a characteristic function of a set. Definition A.2.2. A characteristic function corresponding to a set A ⊂ 𝕄󸀠 is 1A (P)⃗ := {

0 ∞

if P⃗ ∈ A, otherwise.

The second corresponds to the definition of a supergraph. Definition A.2.3. The supergraph of a function Ξ : 𝕄󸀠 → ℝ̂ is the set SG(Ξ) := {(P,⃗ r) ∈ 𝕄󸀠 × ℝ; Ξ(P)⃗ ≥ r}. In particular, SG(Ξ) does not contain the line P⃗ × ℝ whenever Ξ(P)⃗ = ∞. From these definitions we can easily obtain the following. Proposition A.2. 1. A ⊂ P⃗ is a convex set iff 1A is a convex function. 2. P⃗ ∈ A is an extreme point iff it is a strictly convex point of 1A . 3. Ξ is a convex function on 𝕄󸀠 iff SG(Ξ) is a convex set in 𝕄󸀠 × ℝ. ⃗ is an extreme point of SG(Ξ). 4. P⃗ is a strictly convex point of Ξ iff (P,⃗ Ξ(P)) We recall that both convex and closed sets enjoy the property of being preserved under intersections. By the first point of Proposition A.2 and the second point in Proposition A.1 we obtain: Proposition A.3. If {Ξα } is a collection of convex functions, then ⋁α Ξα is a convex function as well. Another nice property of convex sets is the preservation under projection. Let 𝕄 = 𝕄󸀠1 × 𝕄󸀠2 and the projection Proj1 : 𝕄󸀠 → 𝕄󸀠1 is defined as Proj1 (P⃗ 1 , P⃗ 2 ) = P⃗ 1 . One can easily verify that if C ⊂ 𝕄󸀠 is a convex set in 𝕄󸀠 , then Proj1 (C) is convex in 𝕄󸀠1 as well (note that the same statement does not hold for closed sets!). 󸀠

192 | A Convexity Proposition A.4. Let Ξ : 𝕄󸀠1 × 𝕄󸀠1 → ℝ̂ be a convex function. Then Ξ(P⃗ 1 ) := ⋀ Ξ(P⃗ 1 , P⃗ 2 ) P⃗ 2 ∈𝕄󸀠2

is convex on 𝕄󸀠1 . 𝕄󸀠1

Indeed, we observe that SG(Ξ) is the projection from 𝕄󸀠1 × 𝕄󸀠2 × ℝ of SG(Ξ) into × ℝ, and apply Proposition A.2.

A.3 Lower semi-continuity Another closely related notion is lower semi-continuity. Definition A.3.1. A function Ξ is lower semi-continuous (LSC) at a point P⃗ 0 ∈ 𝕄󸀠 iff for any sequence P⃗ n converging to P⃗ 0 lim inf Ξ(P⃗ n ) ≥ Ξ(P⃗ 0 ). n→∞

A function Ξ is said to be LSC if it is LSC at any P⃗ ∈ 𝕄󸀠 . In particular, if Ξ(P⃗ 0 ) = ∞, then Ξ is LSC at P⃗ 0 iff limn→∞ Ξ(P⃗ n ) = ∞ for any sequence P⃗ n → P⃗ 0 . From Definitions A.3.1 and A.2.3 we also get the connection between LSC and closed sets. Proposition A.5. A function Ξ on 𝕄󸀠 is LSC at any point P⃗ ∈ 𝕄󸀠 iff SG(Ξ) is closed in 𝕄󸀠 × ℝ. Warning: Not any convex function is LSC at any point of its essential domain. For example, consider a convex and open set 𝒜 ⊂ 𝕄󸀠 such that any point on the boundary of its closure 𝒜c is an extreme point of 𝒜c (e. g., 𝒜 is the open ball in ℝn ). Let Ξ = 0 on 𝒜, let Ξ = ∞ on ∼ 𝒜c , and let Ξ take arbitrary real values on the boundary of 𝒜. Then Ξ is convex on 𝕄󸀠 and its essential domain is 𝒜c . Still, Ξ is not LSC, in general, at points on the boundary of 𝒜. However, note the following proposition: Proposition A.6. If Ξ is convex on 𝕄󸀠 , then it is continuous at any inner point of its essential domain. Recall that the intersection of a family of closed set is closed as well. Using Propositions A.2, A.5, and A.3 we obtain the following. Proposition A.7. If {Ξβ } is a collection of LSC (resp. convex) functions on 𝕄󸀠 , then Ξ(̄ P)⃗ := ⋁β Ξβ (P)⃗ is an LSC (resp. convex) function as well.

A.4 Legendre transformation

| 193

A.4 Legendre transformation ⃗ Let now {Ξβ } be a collection of affine functions on 𝕄󸀠 , i. e., Ξβ (P)⃗ := γ(β) + P⃗ : Γ(β), ⃗ where γ(β) ∈ ℝ and Γ(β) ∈ 𝕄+ . By Proposition A.7, ⃗ Ξβ (P)⃗ : P⃗ 󳨃→ ⋁ [γ(β) + P⃗ : Γ(β)] ∈ ℝ ∪ {∞} β

is a convex function. ⃗ := In particular, if the set of elements β are points in the dual space 𝕄+ and Γ(M) ⃗ −Ξ(M) is any function on 𝕄+ , then ⃗ := ⋁ [P⃗ : M⃗ − Ξ(P)] ⃗ Σ(M) 󸀠 ⃗ P∈𝕄

(A.1)

is a convex function on 𝕄+ . Thus, the operation (A.1) defines a transformation from the functions on the space 𝕄+ to convex functions on the dual space 𝕄󸀠 . In addition, if we consider only LSC convex functions Ξ in (A.1), it defines the Legendre transform from LSC convex functions on 𝕄+ to LSC convex functions on its dual space 𝕄󸀠 . Since a finite-dimensional linear space is reflexive (i. e., 𝕄󸀠 is the dual of 𝕄+ and 𝕄+ is the dual of 𝕄󸀠 ), we can represent the Legendre transform as a transformation from LSC convex functions Ξ on 𝕄󸀠 to LSC convex functions Ξ∗ on 𝕄+ as well. Definition A.4.1. The Legendre transform of an LSC convex function Ξ on 𝕄󸀠 is the LSC convex function Ξ∗ on 𝕄+ given by ⃗ := ⋁ P⃗ : M⃗ − Ξ(P). ⃗ Ξ∗ (M) 󸀠 ⃗ P∈𝕄

In particular, we obtain ⃗ ≥ P⃗ : M⃗ Ξ(P)⃗ + Ξ∗ (M)

(A.2)

for any P⃗ ∈ 𝕄󸀠 , M⃗ ∈ 𝕄+ . The two-way duality relation between 𝕄󸀠 and 𝕄+ implies the possibility to define ∗∗ Ξ := (Ξ∗ )∗ as an LSC convex function on 𝕄󸀠 . It is an elementary exercise to prove that Ξ∗∗ (P)⃗ ≤ Ξ(P)⃗

(A.3)

for any P⃗ ∈ 𝕄󸀠 . Note that (A.3) holds for any function Ξ : 𝕄󸀠 → ℝ̂ (not necessarily convex or LSC). In fact, for a general function Ξ, Ξ∗∗ is the maximal convex LSC envelop of Ξ, that is, the maximal convex and LSC function dominated by Ξ. However, if Ξ is both convex and LSC on 𝕄󸀠 , then we get an equality in (A.3).

194 | A Convexity Proposition A.8. If Ξ : 𝕄󸀠 → ℝ̂ is convex and LSC on 𝕄󸀠 , then Ξ∗∗ = Ξ. Corollary A.4.1. If Ξ is the support function of a convex closed set A ⊂ 𝕄+ , then its Legendre transform is the characteristic function of A. For the proof of Proposition A.8, see, e. g., [39].

A.5 Subgradients Definition A.5.1. The subgradient of a function Ξ : 𝕄󸀠 → ℝ̂ is defined for any P⃗ in the essential domain of Ξ as 𝜕P⃗ Ξ := {M⃗ ∈ 𝕄+ ; Ξ(P⃗ 1 ) − Ξ(P)⃗ ≥ (P⃗ 1 − P)⃗ : M,⃗ ∀ P⃗ 1 ∈ 𝕄󸀠 } ⊂ 𝕄+ . Note that we only defined 𝜕P⃗ Ξ for P⃗ in the essential domain of Ξ. The reason is to avoid the ambiguous expression ∞ − ∞ in case both Ξ(P)⃗ = Ξ(P⃗ 1 ) = ∞. It can easily be shown that 𝜕P⃗ Ξ is a closed and convex set for any LSC function Ξ. However, it can certainly be the case that the subgradient is an empty set. If, however, Ξ is also convex, then we can guarantee that 𝜕P⃗ Ξ is non-empty. Proposition A.9 ([39]). If Ξ is LSC and convex, then the subgradient 𝜕P⃗ Ξ is non-empty for any P⃗ ∈ Int(ED(Ξ)). If M⃗ ∈ Int(ED(Ξ∗ )), then there exists P⃗ ∈ Int(ED(Ξ)) such that P⃗ ∈ 𝜕M⃗ Ξ∗ and M⃗ ∈ 𝜕P⃗ Ξ. In particular, the equality ⃗ = P⃗ : M⃗ Ξ(P)⃗ + Ξ∗ (M)

(A.4)

holds iff both M⃗ ∈ 𝜕P⃗ Ξ and P⃗ ∈ 𝜕M⃗ Ξ∗ . In particular, P⃗ is a minimizer of Ξ iff 0 ∈ 𝜕P⃗ Ξ (and, of course, M⃗ is a minimizer of Ξ∗ iff 0 ∈ 𝜕M⃗ Ξ∗ ). There is a relation between the differentiability of a convex function and the strict convexity of its Legendre transform. Proposition A.10. An LSC convex function Ξ is differentiable at P⃗ ∈ 𝕄󸀠 iff 𝜕P⃗ Ξ is a singleton, iff its directional derivatives exist on a spanning set of directions. In that case ⃗ Moreover, in that case Ξ∗ is 𝜕P⃗ Ξ is identified with the gradient of Ξ: 𝜕P⃗ Ξ = {∇Ξ(P)}. ⃗ namely, for any α ∈ (0, 1) and any M⃗ 1 ≠ M⃗ 2 verifying strictly convex at M⃗ 0 = ∇Ξ(P), ⃗ ⃗ ⃗ M0 = αM1 + (1 − α)M2 , Ξ∗ (M⃗ 0 ) < αΞ∗ (M⃗ 1 ) + (1 − α)Ξ∗ (M⃗ 2 ). Let us see the proof of the last statement. Let {M⃗ 0 } = 𝜕P⃗ Ξ. Assume there exist α ∈ (0, 1) and M⃗ 1 ≠ M⃗ 2 in the essential domain 0 of Ξ∗ such that M⃗ 0 = αM⃗ 1 + (1 − α)M⃗ 2 and Ξ∗ (M⃗ 0 ) = αΞ∗ (M⃗ 1 ) + (1 − α)Ξ∗ (M⃗ 2 ).

(A.5)

A.6 Support functions | 195

Then from (A.4) Ξ(P⃗ 0 ) − P⃗ 0 : M⃗ 0 = Ξ(P⃗ 0 ) − P⃗ 0 : (αM⃗ 1 + (1 − α)M⃗ 2 ) = −Ξ∗ (M⃗ 0 )

(A.6)

and from (A.2) Ξ(P⃗ 0 ) − P⃗ 0 : M⃗ 1 ≥ −Ξ∗ (M⃗ 1 );

Ξ(P⃗ 0 ) − P⃗ 0 : M⃗ 2 ≥ −Ξ∗ (M⃗ 2 ).

(A.7)

Summing α times the first inequality and (1 − α) times the second inequality of (A.7) we get Ξ(P⃗ 0 ) − P⃗ 0 : (αM⃗ 1 + (1 − α)M⃗ 2 ) ≥ αΞ∗ (M⃗ 1 ) + (1 − α)Ξ∗ (M⃗ 2 ) = Ξ∗ (M⃗ 0 ). From (A.6) we get that the two inequalities in (A.7) are, in fact, equalities: Ξ(P⃗ 0 ) − P⃗ 0 : M⃗ 1 = −Ξ∗ (M⃗ 1 );

Ξ(P⃗ 0 ) − P⃗ 0 : M⃗ 2 = −Ξ∗ (M⃗ 2 ).

Then Proposition A.9 implies that M⃗ 1 , M⃗ 2 ∈ 𝜕P⃗ Ξ. In particular Ξ is not differentiable 0 at P⃗ 0 , in contradiction. Hence (A.5) is violated. Another property of closed convex functions is the following.

Proposition A.11. If {Ξn } is a sequence of LSC convex functions on 𝕄󸀠 and the limit limn→∞ Ξn := Ξ holds pointwise on 𝕄󸀠 , then Ξ is convex and for any interior point P⃗ of the essential domain of Ξ, 𝜕P⃗ Ξ ⊂ lim inf 𝜕P⃗ Ξn . n→∞

It means that for any M⃗ ∈ 𝜕P⃗ Ξ there exists a subsequence M⃗ n ∈ 𝜕P⃗ Ξn converging, as n → ∞, to M.⃗

A.6 Support functions Definition A.6.1. The support function of a set A ⊂ 𝕄+ is defined on the dual space 𝕄󸀠 as SuppA (P)⃗ := ⋁ P⃗ : M.⃗ ⃗ M∈A

In particular, if A is convex and closed, then SuppA is the Legendre transform of the characteristic function of A. Note that the support function is finite everywhere iff A is a compact set. A support function is also positively homogeneous of order 1.

196 | A Convexity Definition A.6.2. A function Ξ on 𝕄󸀠 is positively homogeneous of order 1 if for any real r ≥ 0 and P⃗ ∈ 𝕄󸀠 ⃗ Ξ(r P)⃗ = rΞ(P).

(A.8)

From Proposition A.8 we obtain the following. Proposition A.12. If Ξ is convex, LSC, and positively homogeneous of order 1 on 𝕄󸀠 , then there exists a closed convex set 𝒦 ⊂ 𝕄+ such that Ξ∗ = 1𝒦 on 𝕄+ . In particular, Ξ = Supp𝒦 . ⃗ ≡ Let us sketch the proof of Proposition A.12. Since, in particular, Ξ(0) = 0, Ξ∗ (M) ⃗ ⃗ ⃗ ⃗ supP∈𝕄 󸀠 P : M − Ξ(P) ≥ 0 for any M ∈ 𝕄+ . Moreover, we observe by (A.8) that if ⃗ ⃗ = ∞. Indeed Ξ∗ (M) ⃗ ≥ there exists M⃗ ∈ 𝕄+ for which P⃗ : M⃗ − Ξ(P)⃗ > 0, then Ξ∗ (M) ∗ ⃗ It follows that Ξ is the characteristic function of some 𝒦 ⊂ 𝕄+ . supr≥0 r[P⃗ : M⃗ − Ξ(P)]. Since it is, in addition, a convex and LSC function, it follows from Proposition A.2 that 𝒦 is convex and closed. By Proposition A.8 ∗ ⃗ Ξ(P)⃗ = χ𝒦 (P) ≡ ⋁ P⃗ : M⃗ − χ(P)⃗ = ⋁ P⃗ : M⃗ ⃗ M∈𝕄 +

⃗ M∈𝒦

by the definition of the characteristic function 1𝒦 . From Propositions A.2 and A.10 we also obtain the following. Proposition A.13. P⃗ 0 ≠ 0 is a differentiable point of an LSC, convex, and positively homogeneous of order 1 function Ξ iff M⃗ 0 = ∇Ξ(P⃗ 0 ) is an extreme point of the corresponding closed and convex set 𝒦 satisfying 1∗𝒦 = Ξ.

B Convergence of measures B.1 Total variation A strong notion of convergence of Borel measures on a compact space (X, ℬ) is the convergence in total variations. The total variation (TV) norm is defined by ‖μ1 − μ2 ‖TV =

sup

ϕ∈C(X);|ϕ|≤1

∫ ϕ(dμ1 − dμ2 ).

(B.1)

In fact, the TV norm is taken, in general, as the supremum with respect to the measurable functions bounded by 1. However, in the case of a compact space (or, more generally, in the case of Polish space1 ), the two definitions coincide. In general, this norm is not restricted to probability (or even positive) measures. In particular, the TV distance between a positive measure μ to the zero measure is μ(X). If μ is not a positive measure, then by the Hahn–Jordan decomposition [6] μ = μ+ − μ− , where μ± are both non-negative measures and ‖μ − 0‖TV := ‖μ‖TV = μ+ (X) + μ− (X). In the special case of probability measures, there is another, equivalent definition as follows: ‖μ1 − μ2 ‖TV = 2 sup (μ1 (A) − μ2 (A)) . A∈ℬ

(B.2)

In particular, the TV distance between two probability measures is between 0 and 2. The equivalence between the two definitions (B.1), (B.2) for probability measures is a non-trivial result, based on duality theory (in the spirit of Kantorovich duality mentioned in Section 9.2). The TV norm also induces a notion of distance between measurable sets. Given a positive measure μ on X (e. g., the Lebesgue measure), the TV distance between A, B ∈ ℬ is the TV norm between the measure μ restricted to A and B: ‖A − B‖TV,μ := ‖μ⌊A − μ⌊B‖TV = 2μ(AΔB), where AΔB is the symmetric difference between A and B, namely, AΔB = (A−B)∪(B−A). The reader may compare it with the Hausdorff distance between sets in a metric space (X, d): dH (A, B) := {sup inf d(x, y)} ∨ {sup inf d(x, y)}. x∈A y∈B

1 Separable, completely metrizable topological space. https://doi.org/10.1515/9783110635485-015

x∈B y∈A

198 | B Convergence of measures If μ1 , μ2 are both absolutely continuous with respect to another measure μ, then an equivalent definition (independent of the choice of μ satisfying this condition) is 󵄨󵄨 dμ dμ 󵄨󵄨󵄨 󵄨 ‖μ1 − μ2 ‖TV = ∫󵄨󵄨󵄨 1 − 2 󵄨󵄨󵄨dμ. 󵄨󵄨 dμ dμ 󵄨󵄨

(B.3)

X

The TV norm is, indeed, a strong norm in the sense that it demands a lot from a sequence of measures to converge. Let us consider, for example, the measure μ = δx , where x ∈ X, i. e., the measure defined as δx (A) = {

1 0

if x ∈ A, if x ∈ ̸ A,

∀A ∈ ℬ.

Consider now a sequence xn → x with respect to the topology of X (e. g., limn→∞ d(xn , x) = 0 if (X, d) is a metric space). Then μn := δxn does not converge to δx in the TV norm, unless xn = x for all n large enough. Indeed, one can easily obtain that ‖δx − δy ‖TV = 2 for any x ≠ y.

B.2 Strong convergence The TV norm can be weaken by the following definition. Definition B.2.1. A sequence μn converges strongly to μ if for any A ∈ ℬ lim μn (A) = μ(A). The notion of strong convergence is evidently weaker than TV convergence. Consider, for example, X = [0, 1] and μn (dx) = fn (x)dx, where fn (x) = {

1 0

if ∃k even, x ∈ [k/n, (k + 1)/n), k, otherwise.

Then we can easily verify that μn converges strongly to the uniform measure μ = (1/2)dx on the interval X. However, by (B.3) 1

󵄨󵄨 󵄨 ‖μn − μ‖TV = ∫󵄨󵄨󵄨fn − 󵄨󵄨 0

1 󵄨󵄨󵄨󵄨 󵄨dx → 1/2. 2 󵄨󵄨󵄨

An equivalent definition of strong convergence is the following: μn strongly converges to μ if for any bounded measurable f on X lim ∫ fdμn = ∫ fdμ.

n→∞

X

X

(B.4)

B.3 Weak* convergence

| 199

Indeed, Definition B.2.1 implies this for any characteristic function on ℬ, and hence for any simple function. From here we can extend to any Borel measurable function by a limiting argument. Even though strong convergence is weaker than TV convergence, it is not weak enough. In particular, the sequence δxn does not strongly converge, in general, to δx if x = limn→∞ xn . Indeed, if A = ⋃n {xn } and x ∈ ̸ A, then evidently δxn (A) = 1 for any n but δx (A) = 0. In particular, if, in the above example, xn ≠ xj for n ≠ j, then there is no strongly convergence subsequence of δxn , which, in other words, implies that the strong convergence is not sequentially compact on the set of probability measures.

B.3 Weak* convergence There are many notions of weak* convergence in the literature, which depends on the underlying spaces. Since we concentrate in this book on continuous functions on a compact space, we only need one definition. Let us start with the following observation: Any continuous function is Borel measurable and bounded (due to compactness of X). Therefore, we can integrate any function in C(X) with respect to a given, bounded Borel measure ν ∈ ℳ(X). By the property of integration, this integration we may be viewed as a linear functional on C(X): ν(ϕ) := ∫ ϕdν. X

Definition B.3.1. A sequence of Borel measures {νn } on a compact set X is said to converge weakly* to ν (νn ⇀ ν) if lim ν (ϕ) n→∞ n

= ν(ϕ) ∀ϕ ∈ C(X).

In spite of the apparent similarity between this definition and (B.4), we may observe that this notion of weak* convergence is, indeed, weaker than the strong (and, certainly, TV) convergence. In particular, if νn = δxn and limn→∞ xn = x in X, then νn converges weakly* to δx . Indeed, the continuity of ϕ (in particular, its continuity at the point x ∈ X) implies δxn (ϕ) := ϕ(xn ) → ϕ(x) := δx (ϕ). This is in contrast to strong convergence, as indicated above. The space of continuous functions on a compact set is a Banach space with respect to the supremum norm ‖ϕ‖∞ = sup |ϕ(x)|, x∈X

ϕ ∈ C(X).

200 | B Convergence of measures If we consider C(X), ‖ ⋅ ‖∞ as a Banach space, then any such functional is also continuous |ν(ϕ)| ≤ ν(X)‖ϕ‖∞ . Recall that the set ℳ(X) of bounded Borel measures is also a linear space. We may invert our point of view, and consider any ϕ ∈ C(X) as a linear functional on ℳ(X): ϕ(ν) := ν(ϕ) ∀ν ∈ ℳ(X).

(B.5)

Then, Definition B.3.1 can be understood in the sense that any ϕ ∈ C(X) is a continuous linear functional on ℳ(X), taken with respect to the weak* convergence. Indeed, lim ϕ(νn ) = ϕ(ν)

n→∞

if and only if νn ⇀ ν.

Stated differently, we have the following. The weak* convergence of measures is the weakest topology by which the action (B.5) of any ϕ ∈ C(X ) on ℳ(X ) is continuous.

There is more to say about weak* convergence. The set of all continuous linear functionals on a Banach space B is its dual space, usually denoted by B∗ , which is a Banach space as well with respect to the norm induced by ‖ ⋅ ‖B . Since (C(X), ‖ ⋅ ‖∞ ) is a Banach space, its dual C ∗ (X) contains the space of bounded Borel measures ℳ(X). By the Riesz–Markov–Kakutani representation theorem [27], any continuous functional on (C(K), ‖ ⋅ ‖∞ ) is represented by a finite Borel measure. Thus, C ∗ (X) = ℳ(X).

(B.6)

Now we present the Banach–Alaoglu theorem [40]. Theorem B.1. The closed unit ball of the dual B∗ of a Banach space B (with respect to the norm topology) is compact with respect to the weak* topology. Remark B.3.1. In the case of C ∗ (X), the norm topology is just the TV norm defined in (B.1). Together with (B.6) we obtain the local compactness of ℳ(X) with respect to the weak* topology. There is much more to say about the weak* topology. In particular, the set of probability measures ℳ1 (X) under the weak* topology is metrizable, i. e., there exists a metric on ℳ1 compatible with the weak* topology. This, in fact, is a special case of a general theorem which states that the unit ball of the dual space B∗ of a separable Banach space is metrizable. The interesting part which we stress here is the following

B.3 Weak* convergence

| 201

Theorem B.2. The metric Monge distance, described in Example 9.4.2, is a metrization of the weak* topology on ℳ1 (X). We finish this very fast and dense introduction to weak* convergence by proving this last theorem. Recall (Example 9.4.2) that the metric Monge distance on ℳ1 is given by (9.22) d(μ, ν) = sup ∫ ϕd(ν − μ), ϕ∈Lip(1)

μ, ν ∈ ℳ1 (X).

(B.7)

X

Curiously, this is very similar to the definition of the TV norm (B.1), which is just the norm topology on ℳ1 induced by the supremum norm ‖ ⋅ ‖∞ on C(X). The only difference is that here we consider the supremum on the set of 1-Lipschitz functions, instead of the whole unit ball of (C(X), ‖ ⋅ ‖∞ ). First, we show that a convergence of a sequence νn in the metric Monge distance to ν implies νn ⇀ ν. This follows from the density of Lipschitz functions in (C(X), ‖ ⋅ ‖∞ ). ̃ Given ϕ ∈ C(X) and ϵ > 0, let ϕ̃ ∈ C(X) be a Lipschitz function such that ‖ϕ − ϕ‖ ∞ < ϵ. By the definition of the metric Monge distance, ̃ ̃ ∫ ϕ(dνn − dν) ≤ ϵ + ∫ ϕ(dν n − dν) ≤ ϵ + |ϕ|1 d(νn , ν), X

X

̃ ̃ |ϕ(x)− ϕ(y)| supx=y̸ |x−y|

is the Lipschitz norm of ϕ.̃ where |ϕ|̃ 1 := For the other direction we take advantage of the compactness of the 1-Lipschitz functions in C(X). This implies, in particular, the existence of a maximizer ϕ(ν,μ) in (B.7): d(μ, ν) = ∫ ϕ(μ,ν) d(ν − μ). X

Let now ϕ(νn ,ν) be the sequence of the maximizers realizing d(νn , ν). By the abovementioned compactness, there is a subsequence of the series ϕ(νn ,ν) which converges in k the supremum norm to a function ψ ∈ C(X). Then lim ∫ ψ(dνnk − dν) = 0

k→∞

X

by assumption. It follows that d(νnk , ν) = ∫ ϕ(νn X

k

,ν) (dνnk

− dν) = ∫ ψ(dνnk − dν) + ∫(ϕ(νn X

X

k

,ν)

− ψ)(dνnk − dν).

Since 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨∫(ϕ(νnk ,ν) − ψ)(dνnk − dν)󵄨󵄨󵄨 ≤ ‖ϕ(νnk ,ν) − ψ‖∞ → 0, 󵄨󵄨 󵄨󵄨 󵄨X 󵄨

202 | B Convergence of measures we obtain the convergence of this subsequence to ν in the metric Monge distance. Finally, the same argument implies that any converging subsequence has the same limit ν, thus the whole sequence converges to ν.

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Index Σθ 47 Σθζ 88 +

Σθζ 88 Ξ𝒥 183 Ξ0 12 Ξϵ,+ 62 ζ Ξϵζ 62

Ξθ 150 Ξθ,+ 43, 150 Ξ0,+ 59 ζ Ξ0ζ

59 A⃗ 43 w,ζ 𝒫 ⃗ 57 {M}

𝒪𝒮𝒫 N 39 w,ζ ̄ 𝒮𝒫 ⃗ 57 {M} w,ζ ̄

𝒮𝒫 𝒦

57

Arzelà–Ascoli theorem 124 Birkhoff theorem 9, 51, 52 blocking pairs 1, 3, 17, 20, 23, 36 Borel σ algebra 4, 31, 39 Borel measure XIII, 40, 48, 50, 117, 190, 199 capacity vector 43, 167, 170, 172, 184 cartel 73, 74, 171, 173, 174, 183 characteristic function 48, 72, 191, 194–196, 199 classifier 11, 103–108, 110, 111 coalition ensemble 73, 74, 76, 77, 83, 97, 98, 149 commission 13, 167–170, 174–176, 178 congruent partitions 68, 132, 133, 136–138, 142–146 convex game 180 convex hull 51, 157, 180, 190 cooperative game 2, 13, 22, 176, 178, 182, 183 core 2, 22, 178–180 cyclical monotonicity 2, 37 distortion 108 dominance 65, 67 efficiency 21, 174 escalation 90, 95, 151 essential domain 60, 61, 194, 195

exposed point 189 extreme point 72, 74, 75, 77, 78, 81, 100, 189–192, 196 Gale–Shapley algorithm 3–5, 18, 32 grand coalition 13, 74, 177, 178, 180, 187 Hahn–Banach 86, 100 hedonic market 138, 144 imputations 178, 180 individual (surplus) value 13, 155, 167, 169, 173, 177 information bottleneck XI, 106, 110, 204 Kantorovich duality 118, 120, 155, 157, 197 Krein–Milman theorem 72, 190 Kullback–Leibler divergence 114 Lagrange multiplier 106, 114 Legendre transform 85, 96, 100, 135, 154, 191, 193–195 likelihood 103, 105, 111 Lower semi-continuous (LSC) 120, 192–196 Lyapunov convexity theorem 72 marginal information 104, 106–108, 110, 111 maximal coalition 77, 83, 98 McCann interpolation 128, 141 MinMax theorem 112 Monge 52, 125, 127, 129, 141, 145, 149, 151 Monge distance 127, 145, 201, 202 Nash equilibrium 170, 171, 174, 175, 178 Nash equilibrium (weak) 176, 177 non-transferable marriage 3 optimal partition 4, 9, 15, 100, 102, 105, 117, 139, 150, 154, 157, 158, 161, 162, 165, 185 oversaturated (OS) 40, 41, 57, 99, 117, 149, 152, 161, 162 permutation 8, 28, 51, 52 Polish space 197 Radon–Nikodym derivative 10, 72, 143 saturated 41, 43, 45, 53 semi-finite approximation 132 stable partition 5, 46, 149

208 | Index

strong (deterministic) partition 12, 72, 74, 77, 79, 95, 100, 105, 110, 136, 139 subgradient 93, 97, 194 super additive game 180, 184 superadditivity 180 supergraph 191 supermodularity 180 Support function 195 Total variation (TV) norm 197, 198, 201

unbalanced transport 9, 121 undersaturated (S) 40 undersaturated (US) 41, 43, 45, 53, 57, 58, 99, 117, 149, 152, 159, 161, 164, 168 weak partition 5, 9, 11, 49–53, 66, 68, 83, 103, 107, 112, 113, 118, 133 weak subpartition 57, 66, 84, 88, 100, 118, 179 weak* convergence 48, 72, 133, 189, 199, 200

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